NEET Mock Test 2021

1. The density of material in CGS system of units is 4g/cm³. In a system of units in which unit of length is 10 cm and unit of mass is 100 g, the value of density of material will be

0.4 unit

40 unit

400 unit

0.04 unit

2. The time period of a body under S.H.M. is represented by: T = P^a D^b S^c where P is pressure, D is density and S is surface tension, then values of a, b and c are

$- \frac{3}{2}$ , $\frac{1}{2}$ ,1

-1, - 2, 3

$\frac{1}{2}$ , $- \frac{3}{2}$ , $- \frac{1}{2}$

1, 2, $\frac{1}{3}$

3. The respective number of significant figures for the numbers 23.023, 0.0003 and 2.1 × 10^-3 are

5, 1, 2

5, 1, 5

5, 5, 2

4, 4, 2

4. Young’s modulus of a material has the same unit as that of

pressure

strain

compressibility

force

5. Of the following quantities, which one has dimensions different from the remaining three?

Energy per unit volume

Force per unit area

Product of voltage and charge per unit volume

Angular momentum

6. The pressure on a square plate is measured by measuring the force on the plate and length of the sides of the plate by using the formula $P = \frac{F}{ \ell^{2} }$ . If the maximum errors in the measurement of force and length are 4% and 2% respectively, then the maximum error in the measurement of pressure is

10%

7. The siemen is the SI unit of

resistivity

resistance

conductivity

conductance

8. An object is moving through the liquid. The viscous damping force acting on it is proportional to the velocity. Then dimensions of constant of proportionality are

[ML^-1T^-1]

[MLT^-1]

[M⁰LT^-1]

[ML⁰T^-1]

9. The least count of a stop watch is 0.2 second. The time of 20 oscillations of a pendulum is measured to be 25 second. The percentage error in the measurement of time will be

1.8%

0.8%

0.1%

10. Weber is the unit of

magnetic susceptibility

intensity of magnetisation

magnetic flux

magnetic permeability

11. The physical quantity which has the dimensional [M¹T^-3]formula is

surface tension

solar constant

density

compressibility

12. The dimensions of Wien’s constant are

[ML⁰ T K]

[M⁰ LT⁰ K]

[M⁰ L⁰ T K]

[MLTK]

13. If the capacitance of a nanocapacitor is measured in terms of a unit ‘u’ made by combining the electric charge ‘e’, Bohr radius ‘ $a_{0}$ ’, Planck’s constant ‘h’ and speed of light ‘c’ then

$u = \frac{ e^{2} h }{ a_{0} }$

$u = \frac{hc}{ e^{2} a_{0} }$

$u = \frac{ e^{2} c }{ ha_{0} }$

$u = \frac{ e^{2} a_{0}}{hc}$

14. The dimensions of $\frac{1}{ \in _{0} }$ $\frac{ e^{2} }{hc}$ are

$M^{-1} L^{-3} T^{4} A^{2}$

$ML^{3} T^{-4} A^{-2}$

$M^{0} L^{0} T^{0} A^{0}$

$M^{-1} L^{-3} T^{2} A$

15. The density of a cube is measured by measuring its mass and length of its sides. If the maximum error in the measurement of mass and length are 4% and 3% respectively, the maximum error in the measurement of density will be

12%

13%

16. Which is different from others by units ?

Phase difference

Mechanical equivalent

Loudness of sound

Poisson’s ratio

17. A quantity X is given by $\varepsilon _{0}L \frac{ \Delta V}{ \Delta t}$ where $\in _{0}$ is the permittivity of the free space, L is a length, DV is a potential difference and Dt is a time interval. The dimensional formula for X is the same as that of

resistance

charge

voltage

current

18. If the error in the measurement of the volume of sphere is 6%, then the error in the measurement of its surface area will be

7.5%

19. If velocity (V), force (F) and energy (E) are taken as fundamental units, then dimensional formula for mass will be

$V^{-2} F^{0}E$

$V^{0} FE^{2}$

$VF^{-2} E^{0}$

$V^{-2} F^{0}E$

20. Multiply 107.88 by 0.610 and express the result with correct number of significant figures.

65.8068

65.807

65.81

65.8

21. Which of the following is a dimensional constant?

Refractive index

Poissons ratio

Strain

Gravitational constant

22. If E, m, J and G represent energy, mass, angular momentum and gravitational constant respectively, then the dimensional formula of EJ²/m⁵G² is same as that of the

angle

length

mass

time

23. The refractive index of water measured by the relation $m = \frac {real depth} {apparent depth}$ is found to have values of 1.34, 1.38, 1.32 and 1.36; the mean value of refractive index with percentage error is

1.35 ± 1.48 %

1.35 ± 0 %

1.36 ± 6 %

1.36 ± 0 %

24. If e is the charge, V the potential difference, T the temperature, then the units of $\frac{eV}{T}$ are the same as that of

Planck’s constant

Stefan’s constant

Boltzmann's constant

gravitational constant

25. The dimensions of mobility are

$M^{-2} T^{2}A$

$M^{-1} T^{2}A$

$M^{-2} T^{3}A$

$M^{-1} T^{3}A$

26. Two quantities A and B have different dimensions which mathematical operation given below is physically meaningful?

A/B

A + B

A – B

A = B

27. The velocity of water waves (v) may depend on their wavelength l, the density of water r and the acceleration due to gravity, g. The method of dimensions gives the relation between these quantities is

v $^{2} \propto g \lambda$

v $^{2} \propto g \lambda^{2}$

v $^{2} \propto g^{-1} \lambda^{2}$

28. The physical quantities not having same dimensions are

torque and work

momentum and Planck’s constant

stress and Young’s modulus

speed and $\big( m_{0} e_{0} \big) ^{-1/2}$

29. A physical quantity of the dimensions of length that can be formed out of c, G and $\frac{ e^{2} }{4 \pi \varepsilon_{0} }$ is [c is velocity of light, G is universal constant of gravitation and e is charge]

$c^{2} \begin{bmatrix}G \frac{ e^{2} }{4 \pi \varepsilon_{0} } \end{bmatrix}^{ \frac{1}{2} }$

$\frac{1}{c^{2}} \begin{bmatrix} \frac{ e^{2} }{G4 \pi \varepsilon_{0} } \end{bmatrix}^{ \frac{1}{2} }$

$\frac{1}{c} G \frac{ e^{2} }{4 \pi \varepsilon_{0}}$

$\frac{1}{c^{2}} \begin{bmatrix} G\frac{ e^{2} }{4 \pi \varepsilon_{0} } \end{bmatrix}^{ \frac{1}{2} }$

30. The unit of impulse is the same as that of

energy

power

momentum

velocity

31. If Q denote the charge on the plate of a capacitor of capacitance C then the dimensional formula for $\frac{ Q^{2} }{C}$ is

$\begin{bmatrix} L^{2} M^{2}T \end{bmatrix}$

$\begin{bmatrix} LMT^{2}\end{bmatrix}$

$\begin{bmatrix} L^{2} MT^{-2} \end{bmatrix}$

$\begin{bmatrix} L^{2} M^{2} T^{2} \end{bmatrix}$

32. The mass of the liquid flowing per second per unit area of cross-section of the tube is proportional to (pressure difference across the ends)ⁿand (average velocity of the liquid)^m. Which of the following relations between m and n is correct?

m = n

m = – n

m² = n

m = – n²

33. The Richardson equation is given by I = AT²e^–B/kT. The dimensional formula for AB² is same as that for

I T²

I k²

I k²/T

34. Turpentine oil is flowing through a capillary tube of length $\ell$ and radius r. The pressure difference between the two ends of the tube is p. The viscosity of oil is given by : $\eta = \frac{ p\big( r^{2} - x^{2} \big) }{4v \ell }$ . Here v is velocity of oil at a distance x from the axis of the tube. From this relation, the dimensional formula of $\eta$ is

$\begin{bmatrix}ML^{-1} T^{-1} \end{bmatrix}$

$\begin{bmatrix}MLT^{-1} \end{bmatrix}$

$\begin{bmatrix}ML^{2} T^{-2}\end{bmatrix}$

$\begin{bmatrix}M^{0} L^{0} T^{0}\end{bmatrix}$

35. Given that $y = Asin \begin{bmatrix} \big( \frac{2 \pi }{ \lambda } \big(ct - x\big) \big) \end{bmatrix}$ , where y and x are measured in metre. Which of the following statements is true?

The unit of $\lambda$ is same as that of x and A

The unit of $\lambda$ is same as that of x but not of A

The unit of c is same as that of $\frac{2 \pi }{\lambda}$

The unit of (ct – x) is same as that of $\frac{2 \pi }{\lambda}$

36. If L = 2.331 cm, B = 2.1 cm, then L + B =

4.431 cm

4.43 cm

4.4 cm

4 cm

37. In the relation x = cos ( $\omega$ t + kx), the dimension(s) of $\omega$ is/are

$\begin{bmatrix} M^{0}LT \end{bmatrix}$

$\begin{bmatrix} M^{0} L^{-1} T^{0} \end{bmatrix}$

$\begin{bmatrix} M^{0} L^{0} T^{-1} \end{bmatrix}$

$\begin{bmatrix} M^{0} LT^{-1} \end{bmatrix}$

38. In a vernier callipers, ten smallest divisions of the vernier scale are equal to nine smallest division on the main scale. If the smallest division on the main scale is half millimeter, then the vernier constant is

0.5 mm

0.1 mm

0.05 mm

0.005 mm

39. Which two of the following five physical parameters have the same dimensions?

(A) Energy density (B) Refractive index
(C) Dielectric constant (D) Young’s modulus
(E) Magnetic field

(B) and (D)

(A) and (D)

(A) and (E)

40. In the eqn. $\big(p + \frac{a}{ v^{2} } \big)$ (V - b) = constant, the unit of a is

dyne cm⁵

dyne cm⁴

dyne/cm³

dyne cm²

41. The dimensions of Reynold’s constant are

$\begin{bmatrix} M^{0} L^{0} T^{0} \end{bmatrix}$

$\begin{bmatrix} ML^{-1} T^{-1} \end{bmatrix}$

$\begin{bmatrix} ML^{-1} T^{-2} \end{bmatrix}$

$\begin{bmatrix} ML^{-2} T^{-2} \end{bmatrix}$

42. Which of the following do not have the same dimensional formula as the velocity? Given that $m_{0}$ = permeability of free space, $e_{0}$ = permittivity of free space, n = frequency, l = wavelength, P = pressure, r = density, w = angular frequency, k = wave number,

$1 / \sqrt{ \mu _{0} \varepsilon _{0} }$

n l

$\sqrt{P/ \rho }$

$\omega k$

43. Unit of magnetic moment is

ampere–metre²

ampere–metre

weber–metre²

weber/metre

44. An experiment is performed to obtain the value of acceleration due to gravity g by using a simple pendulum of length L. In this experiment time for 100 oscillations is measured by using a watch of 1 second least count and the value is 90.0 seconds. The length L is measured by using a meter scale of least count 1 mm and the value is 20.0 cm. The error in the determination of g would be:

1.7%

2.7%

4.4%

2.27%

45. The dimensional formula for magnetic flux is

$\begin{bmatrix} ML^{2} T^{-2} A^{-1} \end{bmatrix}$

$\begin{bmatrix} ML^{3} T^{-2} A^{-2} \end{bmatrix}$

$\begin{bmatrix} M^{0} L^{-2} T^{2} A^{-2} \end{bmatrix}$

$\begin{bmatrix} ML^{2} T^{-1} A^{2} \end{bmatrix}$

46. A particle starts moving rectilinearly at time t = 0 such that its velocity v changes with time t according to the equation v = t2 – t where t is in seconds and v is in m/s. Find the time interval for which the particle retards.

$\frac{1}{2}$ < t < 1

$\frac{1}{2}$ > t > 1

$\frac{1}{4}$ < t < 1

$\frac{1}{2}$ < t < $\frac{3}{4}$

47. The co-ordinates of a moving particle at any time ‘t’are given by x = $\alpha t^{3}$ and y = $\beta t^{3}$ . The speed of the particle at time ‘t’ is given by

$3t \sqrt{ \alpha ^{2} + \beta ^{2} }$

$3t^{2} \sqrt{ \alpha ^{2} + \beta ^{2} }$

$t^{2} \sqrt{ \alpha ^{2} + \beta ^{2} }$

$\sqrt{ \alpha ^{2} + \beta ^{2} }$

48. If a car covers 2/5th of the total distance with $v_{1}$ speed and 3/5th distance with $v_{2}$ then average speed is

$\frac{1}{2} \sqrt{ v_{1} v_{2} }$

$\frac{ v_{1} + v_{2} }{2}$

$\frac{ 2v_{1} v_{2} }{ v_{1} + v_{2} }$

$\frac{ 5v_{1} v_{2} } { 3v_{1} + 2v_{2} }$

49. Choose the correct statements from the following.

The magnitude of instantaneous velocity of a particle is equal to its instantaneous speed.

The magnitude of the average velocity in an interval is equal to its average speed in that interval.

It is possible to have a situation in which the speed of the particle is never zero but the average speed in an interval is zero.

It is possible to have a situation in which the speed of particle is zero but the average speed is not zero.

50. A particle located at x = 0 at time t = 0, starts moving along with the positive x-direction with a velocity 'v' that varies as v = $\alpha \sqrt{x}$ . The displacement of the particle varies with time as

t²

$t^{1/2}$

t³

51. Figure here gives the speed-time graph for a body. The displacement travelled between t = 1.0 second and t = 7.0 second is nearest to

1.5 m

2 m

3 m

4 m

52. A particle is moving in a straight line with initial velocity and uniform acceleration a. If the sum of the distance travelled in $t^{th}$ and $(t + 1)^{th}$ seconds is 100 cm, then its velocity after t seconds, in cm/s, is

53. A thief is running away on a straight road on a jeep moving with speed of 9 m/s. A police man chases him on a motor cycle moving at a speed of 10 m/s. If the instantaneous separation of jeep from the motor cycle is 100 m, how long will it take for the police man to catch the thief?

1 second

19 second

90 second

100 second

54. The displacement x of a particle varies with time according to the relation $x = \frac{a}{b} \big(1 -e^{-bt} \big)$ . Then select the false alternative.

At $t = \frac{1}{b}$ , the displacement of the particle is nearly $\frac{2}{3} \big(\frac{a}{b}\big)$

The velocity and acceleration of the particle at t = 0 are a and –ab respectively

The particle cannot go beyond $x = \frac{a}{b}$

The particle will not come back to its starting point at $t \rightarrow \infty$

55. A metro train starts from rest and in five seconds achieves a speed 108 km/h. After that it moves with constant velocity and comes to rest after travelling 45m with uniform retardation. If total distance travelled is 395 m, find total time of travelling.

12.2 s

15.3 s

9 s

17.2 s

56. The deceleration experienced by a moving motor boat after its engine is cut off, is given by dv/dt = – kv $^{3}$ where k is a constant. If v $_{0}$ is the magnitude of the velocity at cut-off, the magnitude of the velocity at a time t after the cut-off is

$\frac{ v_{0} }{ \sqrt{ \big( 2v_{0^{2}kt +1 } \big) } }$

$v_{0} e^{-kt}$

$v_{0}/2$

$v_{0}$

57. The velocity of a particle is $v = v_{0}$ + gt + ft2. If its position is x = 0 at t = 0, then its displacement after unit time (t = 1) is

$v_{0}$ + g /2 + f

$v_{0}$ + 2g + 3f

$v_{0}$ + g /2 + f/3

$v_{0}$ + g + f

58. A man is 45 m behind the bus when the bus starts accelerating from rest with acceleration 2.5 m/s². With what minimum velocity should the man start running to catch the bus?

12 m/s

14 m/s

15 m/s

16 m/s

59. A body is at rest at x = 0. At t = 0, it starts moving in the positive x-direction with a constant acceleration. At the same instant another body passes through x = 0 moving in the positive x-direction with a constant speed. The position of the first body is given by $x_{1}$ (t) after time ‘t’; and that of the second body by $x_{2}$ (t) after the same time interval. Which of the following graphs correctly describes ( $x_{1}$ – $x_{2}$ ) as a function of time ‘t’?

60. From the top of a building 40 m tall, a boy projects a stone vertically upwards with an initial velocity 10 m/s such that it eventually falls to the ground. After how long will the stone strike the ground ? Take g = 10 m/s².

1 s

2 s

3 s

4 s

61. Two bodies begin to fall freely from the same height but the second falls T second after the first. The time (after which the first body begins to fall) when the distance between the
bodies equals L is

$\frac{1}{2}T$

$\frac{T}{2} + \frac{L}{gT}$

$\frac{L}{gT}$

$T+\frac{2L}{gT}$

62. Let A, B, C, D be points on a vertical line such that AB = BC = CD. If a body is released from position A, the times of descent through AB, BC and CD are in the ratio.

$1:\sqrt{3}- \sqrt{2}: \sqrt{3}+ \sqrt{2}$

$1:\sqrt{2}-1:\sqrt{3}-\sqrt{2}$

$1:\sqrt{2}-1:\sqrt{3}$

$1:\sqrt{2}:\sqrt{3}-1$

63. The water drops fall at regular intervals from a tap 5 m above the ground. The third drop is leaving the tap at an instant when the first drop touches the ground. How far above the ground is the second drop at that instant? (Take g = 10 m/s²)

1.25 m

2.50 m

3.75 m

5.00 m

64. The displacement ‘x’ (in meter) of a particle of mass ‘m’ (in kg) moving in one dimension under the action of a force, is related to time ‘t’ (in sec) by t = $\sqrt{x}$ +3. The displacement of the particle when its velocity is zero, will be

2 m

4 m

zero

6 m

65. A body moving with a uniform acceleration crosses a distance of 65 m in the 5 th second and 105 m in 9th second. How far will it go in 20 s?

2040 m

240 m

2400 m

2004 m

66. An automobile travelling with a speed of 60 km/h, can brake to stop within a distance of 20m. If the car is going twice as fast i.e., 120 km/h, the stopping distance will be

60 m

40 m

20 m

80 m

67. A particle accelerates from rest at a constant rate for some time and attains a velocity of 8 m/sec. Afterwards it decelerates with the constant rate and comes to rest. If the total time taken is 4 sec, the distance travelled is

32 m

16 m

4 m

None of the above

68. The equation represented by the graph below is :

y = $\frac{1}{2} gt$

y = $\frac{-1}{2} gt$

y = $\frac{1}{2} gt^{2}$

y = $\frac{-1}{2} gt^{2}$

69. A particle moves a distance x in time t according to equation x = (t + 5)^–1. The acceleration of particle is proportional to:

(velocity)^3/2

(distance)²

(distance)^-2

(velocity)^2/3

70. A particle when thrown, moves such that it passes from same height at 2 and 10 seconds, then this height h is :

10g

71. The distance through which a body falls in the nth second is h. The distance through which it falls in the next second is

h + $\frac{g}{2}$

h – g

h + g

72. A stone thrown upward with a speed u from the top of the tower reaches the ground with a velocity 3u. The height of the tower is

3u²/g

4u²/g

6u²/g

9u²/g

73. A particle moves along a straight line OX. At a time t (in seconds) the distance x (in metres) of the particle from O is given by x = 40 + 12t – t³. How long would the particle travel
before coming to rest?

40 m

56 m

16 m

24 m

74. The graph shown in figure shows the velocity v versus time t for a body. Which of the graphs represents the corresponding acceleration versus time graphs?

75. A particle moving along x-axis has acceleration f, at time t, given by $f = f_{0} \big(1- \frac{t}{T} \big)$ where $f_{0}$ and T are constants. The particle at t = 0 has zero velocity. In the time interval between t = 0 and the instant when f = 0, the particle’s velocity ( $v_{x}$ ) is

$\frac{1}{2} f_{0} T^{2}$

$f_{0} T^{2}$

$\frac{1}{2} f_{0}T$

$f_{0}T$

76. A body is thrown vertically up with a velocity u. It passes three points A, B and C in its upward journey with velocities $\frac{u}{2}$ , $\frac{u}{3}$ and $\frac{u}{4}$ respectively. The ratio of AB and BC is

20 : 7

10 : 7

77. A boat takes 2 hours to travel 8 km and back in still water lake. With water velocity of 4 km h^–1, the time taken for going upstream of 8 km and coming back is

160 minutes

80 minutes

100 minutes

120 minutes

78. A body starts from rest and travels a distance x with uniform acceleration, then it travels a distance 2x with uniform speed, finally it travels a distance 3x with uniform retardation and comes to rest. If the complete motion of the particle is along a straight line, then the ratio of its average velocity to maximum velocity is

2/5

3/5

4/5

6/7

79. A man of 50 kg mass is standing in a gravity free space at a height of 10 m above the floor. He throws a stone of 0.5 kg mass downwards with a speed 2 m/s. When the stone reaches the floor, the distance of the man above the floor will be:

9.9 m

10.1 m

10 m

20 m

80. A boy moving with a velocity of 20 km h^–1 along a straight line joining two stationary objects. According to him both objects

move in the same direction with the same speed of 20 km h^–1

move in different direction with the same speed of 20 km h^–1

move towards him

remain stationary

81. A rubber ball is dropped from a height of 5 metre on a plane where the acceleration due to gravity is same as that onto the surface of the earth. On bouncing, it rises to a height of 1.8 m. On bouncing, the ball loses its velocity by a factor of

$\frac{3}{5}$

$\frac{9}{25}$

$\frac{2}{5}$

$\frac{16}{25}$

82. A stone falls freely from rest from a height h and it travels a distance $\frac{9h}{25}$ in the last second. The value of h is

145 m

100 m

122.5 m

200 m

83. Which one of the following equations represents the motion of a body with finite constant acceleration ? In these equations, y denotes the displacement of the body at time t and a, b and c are constants of motion.

y = at

y = at + bt²

y = at + bt² + ct³

y = $\frac{a}{t} + bt$

84. A particle travels half the distance with a velocity of 6 ms^–1. The remaining half distance is covered with a velocity of 4 ms^–1 for half the time and with a velocity of 8 ms^–1 for the rest of the half time. What is the velocity of the particle averaged over the whole time of motion ?

9 ms^–1

6 ms^–1

5.35 ms^–1

5 ms^–1

85. A bullet is fired with a speed of 1000 m/sec in order to penetrate a target situated at 100 m away. If g = 10 m/s², the gun should be aimed

directly towards the target

5 cm above the target

10 cm above the target

15 cm above the target

86. A body covers 26, 28, 30, 32 meters in 10th, 11th, 12th and 13th seconds respectively. The body starts

from rest and moves with uniform velocity

from rest and moves with uniform acceleration

with an initial velocity and moves with uniform acceleration

with an initial velocity and moves with uniform velocity

87. A particle is moving with uniform acceleration along a straight line. The average velocity of the particle from P to Q is 8ms^–1 and that Q to S is 12ms^–1. If QS = PQ, then the average velocity from P to S is

9.6 ms^–1

12.87 ms^–1

64 ms^–1

327 ms^–1

88. The variation of velocity of a particle with time moving along a straight line is illustrated in the figure. The distance travelled by the particle in four seconds is

60 m

55 m

25 m

30 m

89. A stone falls freely under gravity. It covers distances $h_{1}$ , $h_{2}$ and $h_{3}$ in the first 5 seconds, the next 5 seconds and the next 5 seconds respectively. The relation between $h_{1}$ , $h_{2}$ and $h_{3}$ is

$h_{1} = \frac{ h_{2} }{3} = \frac{ h_{3} }{5}$

$h_{2} = 3h_{1}$ and $h_{3} = 3h_{2}$

$h_{1} = h_{2} = h_{3}$

$h_{1} = 2h_{2} = 3h_{3}$

90. A car, starting from rest, accelerates at the rate f through a distance S, then continues at constant speed for time t and then decelerates at the rate $\frac{f}{2}$ to come to rest. If the total distance traversed is 15 S , then

$S = \frac{1}{6} ft^{2}$

S = f t

$S = \frac{1}{4} ft^{2}$

$S = \frac{1}{72} ft^{2}$

91. A projectile is given an initial velocity of $\big(\hat{i} + 2\hat{j}\big)$ m/s, where $\hat{i}$ is along the ground and $\hat{j}$ is along the vertical. If g = 10 m/s² , the equation of its trajectory is :

y = x – 5x²

y = 2x – 5x²

4y = 2x – 5x²

4y = 2x – 25x²

92. An aircraft moving with a speed of 250 m/s is at a height of 6000 m, just overhead of an anti aircraft–gun. If the muzzle velocity is 500 m/s, the firing angle $\theta$ should be:

30°

45°

60°

75°

93. Two racing cars of masses $m_{1}$ and $m_{2}$ are moving in circles of radii $r_{1}$ and $r_{2}$ respectively. Their speeds are such that each makes a complete circle in the same duration of time t. The ratio of the angular speed of the first to the second car is

$m_{1} : m_{2}$

$r_{1} : r_{2}$

1 : 1

$m_{1} r_{1} : m_{2}r_{2}$

94. A boy playing on the roof of a 10 m high building throws a ball with a speed of 10m/s at an angle of 30º with the horizontal. How far from the throwing point will the ball be at the height of 10 m from the ground ?

[ g = 10m/s², sin30º = $\frac{1}{2}$ , cos30º = $\frac{ \sqrt{3} }{2}$ ]

5.20m

4.33m

2.60m

8.66m

95. A bomber plane moves horizontally with a speed of 500 m/s and a bomb released from it, strikes the ground in 10 sec. Angle at which it strikes the ground wil be (g = 10 m/s²)

$tan^{-1} \big(\frac{1}{5}\big)$

$tan \big(\frac{1}{5}\big)$

$tan^{-1} \big(1\big)$

$tan^{-1} \big(5\big)$

96. Two particles start simultaneously from the same point and move along two straight lines, one with uniform velocity v and other with a uniform acceleration a. If $\alpha$ is the angle between the lines of motion of two particles then the least value of relative velocity will be at time given by

$\frac{v}{a} sin\alpha$

$\frac{v}{a} cos\alpha$

$\frac{v}{a} tan\alpha$

$\frac{v}{a} cot\alpha$

97. Initial velocity with which a body is projected is 10 m/sec and angle of projection is 60°. Find the range R

$\frac{15 \sqrt{3}m }{2}$

$\frac{40}{3}m$

$5\sqrt{3}m$

$\frac{20}{3}m$

98. The position vectors of points A, B, C and D are $A = 3 \hat{i} + 4\hat{j} + 5 \hat{k}, B = 4 \hat{i} + 5 \hat{j} + 6 \hat{k}, C = 7 \hat{i} + 9 \hat{j} + 3 \hat{k}$ and $D = 4 \hat{i} + 6 \hat{j}$ then the displacement vectors $\overline{AB}$ and $\overline{CD}$ are

perpendicular

parallel

antiparallel

inclined at an angle of 60°

99. A person swims in a river aiming to reach exactly on the opposite point on the bank of a river. His speed of swimming is 0.5 m/s at an angle of 120º with the direction of flow of water. The speed of water is

1.0 m/s

0.5 m/s

0.25 m/s

0.43 m/s

100. A projectile thrown with velocity v making angle $\theta$ with vertical gains maximum height H in the time for which the projectile remains in air, the time period is

$\sqrt{H cos\theta/g}$

$\sqrt{2H cos\theta/g}$

$\sqrt{4H/g}$

$\sqrt{8H/g}$

101. A ball is thrown from a point with a speed ' $v_{0}$ ' at an elevation angle of $\theta$ . From the same point and at the same instant, a person starts running with a constant speed $\frac{'v_{0}}{2}'$ to catch the ball. Will the person be able to catch the ball? If yes, what should be the angle of projection $\theta$ ?

No, 0°

Yes, 30°

Yes, 60°

Yes, 45°

102. If vectors $\vec{A} = cos\omega t\hat{i} + sin\omega t\hat{j}$ and $\vec{B} = cos\frac{ \omega t}{2} \hat{i} + sin\frac{ \omega t}{2} \hat{j}$ are functions of time, then the value of t at which they are orthogonal to each other is :

$t = \frac{ \pi }{ 2\omega }$

$t = \frac{ \pi }{ \omega }$

t = 0

$t = \frac{ \pi }{ 4\omega }$

103. A bus is moving on a straight road towards north with a uniform speed of 50 km/hour turns through 90°. If the speed remains unchanged after turning, the increase in the velocity of bus in the turning process is

70.7 km/hour along south-west direction

70.7 km/hour along north-west direction.

50 km/hour along west

zero

104. The velocity of projection of oblique projectile is $\big( 6\hat{i} + 8\hat{j} \big) ms^{-1}$ . The horizontal range of the projectile is

4.9 m

9.6 m

19.6 m

14 m

105. A point P moves in counter-clockwise direction on a circular path as shown in the figure. The movement of 'P' is such that it sweeps out a length s = t³ + 5, where s is in metres and t is in seconds. The radius of the path is 20 m. The acceleration of 'P' when t = 2 s is nearly

13 m/s²

12 m/s²

7.2 ms²

14 m/s²

106. The resultant of two vectors $\vec{A}$ and $\vec{B}$ is perpendicular to the vector $\vec{A}$ and its magnitude is equal to half the magnitude of vector $\vec{B}$ The angle between $\vec{A}$ and $\vec{B}$ is

120°

150°

135°

180°

107. A man running along a straight road with uniform velocity $\vec{u} = u \hat{i}$ feels that the rain is falling vertically down along $-\hat{j}$ . If he doubles his speed, he finds that the rain is coming at an angle $\theta$ with the vertical. The velocity of the rain with respect to the ground is

ui – uj

$ui - \frac{u}{tan\theta} \hat{j}$

$2u\hat{i} + u cot \theta \hat{j}$

$ui + u sin \theta \hat{j}$

108. Two projectiles A and B thrown with speeds in the ratio $1 : \sqrt{2}$ acquired the same heights. If A is thrown at an angle of 45° with the horizontal, the angle of projection of B will be

0°

60°

30°

45°

109. A projectile can have the same range ‘R’ for two angles of projection. If ‘T1’ and ‘T2’ be time of flights in the two cases, then the product of the two time of flights is directly proportional to

$\frac{1}{R}$

$\frac{1}{R^{2} }$

${R^{2} }$

110. A man standing on the roof of a house of height h throws one particle vertically downwards and another particle horizontally with the same velocity u. The ratio of their velocities when they reach the earth's surface will be

$\sqrt{2gh + u^{2} }:u$

1 : 2

1 : 1

$\sqrt{2gh + u^{2} }: \sqrt{2gh}$

111. If a unit vector is represented by $0.5\hat{i} + 0.8\hat{j} + c\hat{k}$ , the value of c is

$\sqrt{0.11}$

$\sqrt{0.01}$

0.39

112. An aeroplane is flying at a constant horizontal velocity of 600 km/hr at an elevation of 6 km towards a point directly above the target on the earth's surface. At an appropriate time, the pilot releases a ball so that it strikes the target at the earth. The ball will appear to be falling

on a parabolic path as seen by pilot in the plane

vertically along a straight path as seen by an observer on the ground near the target

on a parabolic path as seen by an observer on the ground near the target

on a zig-zag path as seen by pilot in the plane

113. A particle is projected with a velocity v such that its range on the horizontal plane is twice the greatest height attained by it. The range of the projectile is (where g is acceleration due to gravity)

$\frac{4v^{2} }{5g}$

$\frac{4g}{5v^{2} }$

$\frac{v^{2} }{g}$

$\frac{4v^{2} }{ \sqrt{5g} }$

114. Two stones are projected from the same point with same speed making angles (45° + $\theta$ ) and (45° – $\theta$ ) with the horizontal respectively. If $\theta \leq$ 45° , then the horizontal ranges of the two stones are in the ratio of

1 : 1

1 : 2

1 : 3

1 : 4

115. Three forces acting on a body are shown in the figure. To have the resultant force only along the y-direction, the magnitude of the minimum additional force needed is:

0.5 N

1.5 N

$\frac{\sqrt{3}}{4}N$

$\sqrt{3}N$

116. A particle moves in x-y plane under the action of force $\vec{F}$ and $\vec{p}$ at a given time t $p_{x}$ = 2 cos $\theta$ , $p_{y}$ = 2sin $\theta$ .Then the angle $\theta$ between $\vec{F}$ and $\vec{p}$ at a given time t is :

$\theta$ = 30°

$\theta$ = 180°

$\theta$ = 0°

$\theta$ = 90°

117. A person sitting in the rear end of the compartment throws a ball towards the front end. The ball follows a parabolic path. The train is moving with velocity of 20 m/s. A person standing outside on the ground also observes the ball. How will the maximum heights ( $y_{m}$ ) attained and the ranges (R) seen by the thrower and the outside observer compare with each other?

Same $y_{m}$ different R

Same $y_{m}$ and R

Different $y_{m}$ same R

Different $y_{m}$ and R

118. A car moves on a circular road. It describes equal angles about the centre in equal intervals of time. Which of the following statement about the velocity of the car is true ?

Magnitude of velocity is not constant

Both magnitude and direction of velocity change

Velocity is directed towards the centre of the circle

Magnitude of velocity is constant but direction changes

119. Three particles A, B and C are thrown from the top of a tower with the same speed. A is thrown up, B is thrown down and C is horizontally. They hit the ground with speeds $v_{A}$ , $v_{B}$ and $v_{C}$ respectively then,

$v_{A} = v_{B} = v_{C}$

$v_{A} = v_{B} > v_{C}$

$v_{B} > v_{C} > v_{A}$

$v_{A} > v_{B} = v_{C}$

120. A particle is moving such that its position coordinate (x, y) are
(2m, 3m) at time t = 0
(6m, 7m) at time t = 2 s and
(13m, 14m) at time t = 5s.
Average velocity vector $\big(\vec{V}_{av}\big)$ from t = 0 to t = 5s is :

$\frac{1}{5} \big(13\hat{i} + 14\hat{j}\big)$

$\frac{7}{3} \big(\hat{i} + \hat{j}\big)$

$2 \big(\hat{i} + \hat{j}\big)$

$\frac{11}{5} \big(\hat{i} + \hat{j}\big)$

121. A particle moves so that its position vector is given by $\vec{r} = cos\omega t \hat{x} + sin\omega t \hat{y}$ . Where $\omega$ is a constant. Which of the following is true ?

Velocity and acceleration both are perpendicular to $\vec{r}$

Velocity and acceleration both are parallel to $\vec{r}$

Velocity is perpendicular to $\vec{r}$ and acceleration is directed towards the origin

Velocity is perpendicular to $\vec{r}$ and acceleration is directed away from the origin

122. Two boys are standing at the ends A and B of a ground where AB = a. The boy at B starts running in a direction perpendicular to AB with velocity $v_{1}$ . The boy at A starts running simultaneously with velocity v and catches the other boy in a time t, where t is

$a/ \sqrt{ v^{2} + v_{1}^{2} }$

$a/\big( v + v_{1} \big)$

$a/\big( v - v_{1} \big)$

$\sqrt{{ a^{2}/ \big(v^{2} - v_{1}^{2}\big) }}$

123. A projectile is fired at an angle of 45° with the horizontal. Elevation angle of the projectile at its highest point as seen from the point of projection is

60°

$tan^{-1} \big( \frac{1}{2} \big)$

$tan^{-1} \big( \frac{ \sqrt{3} }{2} \big)$

45°

124. The position vector of a particle $\vec{R}$ as a function of time is given by $\vec{R} = 4sin \big(2 \pi t\big) \hat{i} + 4cos \big(2 \pi t\big) \hat{j}$
where R is in meter, t in seconds and $\hat{i}$ and $\hat{j}$ denote unit vectors along x-and y-directions, respectively. Which one of the following statements is wrong for the motion of particle?

Magnitude of acceleration vector is $\frac{ v^{2} }{R}$ , where v is the velocity of particle

Magnitude of the velocity of particle is 8 meter/second

Path of the particle is a circle of radius 4 meter.

Acceleration vector is along - $\vec{R}$

125. The vectors $\vec{A}$ and $\vec{B}$ are such that | $\vec{A} + \vec{B}$ |=| $\vec{A} - \vec{B}$ | The angle between the two vectors is

60°

75°

45°

90°

126. The velocity of projection of oblique projectile is $\big( 6\hat{i}+8\hat{j} \big)ms^{-1}$ . The horizontal range of the projectile is

4.9 m

9.6 m

19.6 m

14 m

127. An artillary piece which consistently shoots its shells with the same muzzle speed has a maximum range R. To hit a target which is R/2 from the gun and on the same level, the elevation angle of the gun should be

15°

45°

30°

60°

128. A car runs at a constant speed on a circular track of radius 100 m, taking 62.8 seconds in every circular loop. The average velocity and average speed for each circular loop respectively, is

0, 10 m/s

10 m/s, 10 m/s

10 m/s, 0

0, 0

129. A vector of magnitude b is rotated through angle $\theta$ . What is the change in magnitude of the vector?

2b $sin\frac{ \theta }{2}$

2b $cos\frac{ \theta }{2}$

2b sin $\theta$

2b cos $\theta$

130. A stone projected with a velocity u at an angle $\theta$ with the horizontal reaches maximum height $H_{1}$ . When it is projected with velocity u at an angle $\big(\frac{ \pi }{2} -\theta\big)$ with the horizontal, it reaches maximum height $H_{2}$ . The relation between the horizontal range R of the projectile, heights $H_{1}$ and $H_{2}$ is

$R = 4\sqrt{ H_{1}H_{2}}$

$R = 4 \big( H_{1} - H_{2}\big)$

$R = 4 \big( H_{1} + H_{2}\big)$

$R = \frac{ H^{2}_{1} }{H^{2}_{2}}$

131. The vector sum of two forces is perpendicular to their vector differences. In that case, the forces

cannot be predicted

are equal to each other

are equal to each other in magnitude

are not equal to each other in magnitude

132. A particle crossing the origin of co-ordinates at time t = 0, moves in the xy-plane with a constant acceleration a in the y-direction. If its equation of motion is y = bx² (b is a constant), its velocity component in the x-direction is

$\sqrt{ \frac{2b}{a} }$

$\sqrt{ \frac{a}{2b} }$

$\sqrt{ \frac{a}{b} }$

$\sqrt{ \frac{b}{a} }$

133. A vector $\vec{A}$ is rotated by a small angle $\Delta \theta$ radian ( $\Delta \theta$ <<1) to get a new vector $\vec{B}$ In that case | $\vec{B}$ - $\vec{A}$ | is :

| $\vec{A}$ | $\Delta \theta$

| $\vec{B}$ | $\Delta \theta$ -| $\vec{A}$ |

| $\vec{A}$ | $\big(1- \frac{\Delta \theta^{2}}{2} \big)$

134. If a body moving in circular path maintains constant speed of 10 ms^–1, then which of the following correctly describes relation between acceleration and radius ?

135. The position of a projectile launched from the origin at t = 0 is given by $\vec{r} = \big(40\hat{i} + 50\hat{j} \big)m$ at t = 2s. If the projectile was launched at an angle $\theta$ from the horizontal, then $\theta$ is (take g = 10 ms^–2)

$tan^{-1} \frac{2}{3}$

$tan^{-1} \frac{3}{2}$

$tan^{-1} \frac{7}{4}$

$tan^{-1} \frac{4}{5}$

136. A player stops a football weighing 0.5 kg which comes flying towards him with a velocity of 10m/s. If the impact lasts for 1/50th sec. and the ball bounces back with a velocity of 15 m/s, then the average force involved is

250 N

1250 N

500 N

625 N

137. For the given situation as shown in the figure, the value of $\theta$ to keep the system in equilibrium will be

30°

45°

0°

90°

138. A 5000 kg rocket is set for vertical firing. The exhaust speed is 800 m/s. To give an initial upward acceleration of 20 m/s², the amount of gas ejected per second to supply the needed thrust will be (Take g = 10 m/s²)

127.5 kg/s

137.5 kg/s

155.5 kg/s

187.5 kg/s

139. Which one of the following statements is correct?

If there were no friction, work need to be done to move a body up an inclined plane is zero.

If there were no friction, moving vehicles could not be stopped even by locking the brakes.

As the angle of inclination is increased, the normal reaction on the body placed on it increases.

A duster weighing 0.5 kg is pressed against a vertical board with force of 11 N. If the coefficient of friction is 0.5, the work done in rubbing it upward through a distance of 10 cm is 0.55 J.

140. A stone is dropped from a height h. It hits the ground with a certain momentum P. If the same stone is dropped from a height 100% more than the previous height, the momentum when it hits the ground will change by :

68%

41%

200%

100%

141. A 3 kg ball strikes a heavy rigid wall with a speed of 10 m/s at an angle of 60º. It gets reflected with the same speed and angle as shown here. If the ball is in contact with the wall for 0.20s, what is the average force exerted on the ball by the wall?

150 N

zero

$150\sqrt{3}N$

300 N

142. The upper half of an inclined plane of inclination $\theta$ is perfectly smooth while lower half is rough. A block starting from rest at the top of the plane will again come to rest at the bottom, if the coefficient of friction between the block and lower half of the plane is given by

$\mu = \frac{2}{tan \theta}$

$\mu = 2 tan\theta$

$\mu = tan\theta$

$\mu = \frac{1}{tan \theta}$

143. A block of mass m is in contact with the cart C as shown in the figure.

The coefficient of static friction between the block and the cart is $\mu$ . The acceleration $\alpha$ of the cart that will prevent the block from falling satisfies:

$\alpha > \frac{mg}{\mu}$

$\alpha > \frac{g}{\mu m}$

$\alpha \geq \frac{g}{\mu}$

$\alpha < \frac{g}{\mu}$

144. A bridge is in the from of a semi-circle of radius 40m. The greatest speed with which a motor cycle can cross the bridge without leaving the ground at the highest point is (g = 10 m s^-2) (frictional force is negligibly small)

40 m s^-1

20 m s^-1

30 m s^-1

15 m s^-1

145. An explosion blows a rock into three parts. Two parts go off at right angles to each other. These two are, 1 kg first part moving with a velocity of 12 ms^-1 and 2 kg second part moving with a velocity of 8 ms^-1. If the third part flies off with a velocity of 4 ms^-1, its mass would be

5 kg

7 kg

17 kg

3 kg

146. A monkey is descending from the branch of a tree with constant acceleration. If the breaking strength is 75% of the weight of the monkey, the minimum acceleration with which monkey can slide down without breaking the branch is

$\frac {3g}{4}$

$\frac {g}{4}$

$\frac {g}{2}$

147. A car having a mass of 1000 kg is moving at a speed of 30 metres/sec. Brakes are applied to bring the car to rest. If the frictional force between the tyres and the road surface is 5000 newtons, the car will come to rest in

5 seconds

10 seconds

12 seconds

6 seconds

148. A spring is compressed between two toy carts of mass $m_{1}$ and $m_{2}$ . When the toy carts are released, the springs exert equal and opposite average forces for the same time on each toy cart. If $v_{1}$ and $v_{2}$ are the velocities of the toy carts and there is no friction between the toy carts and the ground, then :

v1/v2 = m1/m2

v1/v2 = m2/m1

v1/v2 = –m2/m1

v1/v2 = –m1/m2

149. A plate of mass M is placed on a horizontal frictionless surface (see figure), and a body of mass m is placed on this plate. The coefficient of dynamic friction between this body and the plate is $\mu$ . If a force 2 $\mu$ mg is applied to the body of mass m along the horizontal, the acceleration of the plate will be

$\frac{\mu m}{M}g$

$\frac{\mu m}{M + m}g$

$\frac{2\mu m}{M}g$

$\frac{2\mu m}{M + m}g$

150. The rate of mass of the gas emitted from rear of a rocket is initially 0.1 kg/sec. If the speed of the gas relative to the rocket is 50 m/sec and mass of the rocket is 2 kg, then the acceleration of the rocket in m/sec² is

5.2

2.5

151. A plank with a box on it at one end is gradually raised about the other end. As the angle of inclination with the horizontal reaches 30º the box starts to slip and slides 4.0 m down the plank in 4.0s. The coefficients of static and kinetic friction between the box and the plank will be, respectively :

0.6 and 0.5

0.5 and 0.6

0.4 and 0.3

0.6 and 0.6

152. Four blocks of same mass connected by cords are pulled by a force F on a smooth horizontal surface, as shown in fig. The tensions $T_{1}$ , $T_{2}$ and $T_{3}$ will be

$T_{1} = \frac{1}{4}F, T_{2} = \frac{3}{2}F, T_{3} = \frac{1}{4}F$

$T_{1} = \frac{1}{4}F, T_{2} = \frac{1}{2}F, T_{3} = \frac{1}{2}F$

$T_{1} = \frac{3}{4}F, T_{2} = \frac{1}{2}F, T_{3} = \frac{1}{4}F$

$T_{1} = \frac{3}{4}F, T_{2} = \frac{1}{2}F, T_{3} = \frac{1}{2}F$

153. A body of mass M is kept on a rough horizontal surface (friction coefficient μ). A person is trying to pull the body by applying a horizontal force but the body is not moving. The force by the surface on the body is F, then

F = Mg

F = μMg

$Mg \leq F \leq Mg \sqrt{1 + \mu^{2} }$

$Mg \geq F \geq Mg \sqrt{1 + \mu^{2} }$

154. Which one of the following motions on a smooth plane surface does not involve force?

Accelerated motion in a straight line

Retarded motion in a straight line

Motion with constant momentum along a straight line

Motion along a straight line with varying velocity

155. A block A of mass $m_{1}$ rests on a horizontal table. A light string connected to it passes over a frictionless pulley at the edge of table and from its other end another block B of mass $m_{2}$ is suspended. The coefficient of kinetic friction between the block and the table is $\mu_{k}$ . When the block A is sliding on the table, the tension in the string is

$\frac{\big(m_{2} - \mu_{k} m_{1}\big)g}{ \big( m_{1} + m_{2}\big) }$

$\frac{ m_{1}m_{2}\big(1 + \mu_{k}\big)g}{ \big( m_{1} + m_{2}\big) }$

$\frac{ m_{1}m_{2}\big(1 - \mu_{k}\big)g}{ \big( m_{1} + m_{2}\big) }$

$\frac{\big(m_{2} + \mu_{k} m_{1}\big)g}{ \big( m_{1} + m_{2}\big) }$

156. The upper half of an inclined plane with inclination f is perfectly smooth while the lower half is rough. A body starting from rest at the top will again come to rest at the bottom if the coefficient of friction for the lower half is given by

2 cos $\phi$

2 sin $\phi$

tan $\phi$

2 tan $\phi$

157. A particle describes a horizontal circle in a conical funnel whose inner surface is smooth with speed of 0.5 m/s. What is the height of the plane of circle from vertex of the funnel?

0.25 cm

2 cm

4 cm

2.5 cm

158. You are on a friction less horizontal plane. How can you get off if no horizontal force is exerted by pushing against the surface?

By jumping

By spitting or sneezing

by rolling your body on the surface

By running on the plane

159. The coefficient of static and dynamic friction between a body and the surface are 0.75 and 0.5 respectively. A force is applied to the body to make it just slide with a constant acceleration which is

$\frac{g}{4}$

$\frac{g}{2}$

$\frac{3g}{2}$

160. In the system shown in figure, the pulley is smooth and massless, the string has a total mass 5g, and the two suspended blocks have masses 25 g and 15 g. The system is released from state $\ell$ = 0 and is studied upto stage $\ell'$ = 0. During the process, the acceleration of block A will be

constant at $\frac{g}{9}$

constant at $\frac{g}{4}$

increasing by factor of 3

increasing by factor of 2

161. The minimum force required to start pushing a body up rough (frictional coefficient $\mu$ ) inclined plane is $F_{1}$ while the minimum force needed to prevent it from sliding down is $F_{2}$ . If the inclined plane makes an angle $\theta$ from the horizontal such that tan $\theta$ = 2 $\mu$ then the ratio $\frac{ F_{1} }{ F_{2}}$ is

162. Two blocks are connected over a massless pulley as shown in fig. The mass of block A is 10 kg and the coefficient of kinetic friction is 0.2. Block A slides down the incline at constant speed. The mass of block B in kg is

3.5

3.3

3.0

2.5

163. Tension in the cable supporting an elevator, is equal to the weight of the elevator. From this, we can conclude that the elevator is going up or down with a

uniform velocity

uniform acceleration

variable acceleration

either (b) or (c)

164. A particle tied to a string describes a vertical circular motion of radius r continually. If it has a velocity $\sqrt{3 gr}$ at the highest point, then the ratio of the respective tensions in the string holding it at the highest and lowest points is

4 : 3

5 : 4

1 : 4

3 : 2

165. It is difficult to move a cycle with brakes on because

rolling friction opposes motion on road

sliding friction opposes motion on road

rolling friction is more than sliding friction

sliding friction is more than rolling friction

166. A plumb line is suspended from a celling of a car moving with horizontal acceleration of a. What will be the angle of inclination with vertical?

$tan^{-1}$ (a/g)

$tan^{-1}$ (g/a)

$cos^{-1}$ (a/g)

$cos^{-1}$ (g/a)

167. A cart of mass M has a block of mass m attached to it as shown in fig. The coefficient of friction between the block and the cart is μ. What is the minimum acceleration of the cart so that the block m does not fall?

$\mu g$

$g/\mu$

$\mu/g$

$M \mu g/m$

168. What is the maximum value of the force F such that the block shown in the arrangement, does not move?

20 N

10 N

12 N

15 N

169. A block has been placed on an inclined plane with the slope angle $\theta$ , block slides down the plane at constant speed. The coefficient of kinetic friction is equal to

sin $\theta$

cos $\theta$

tan $\theta$

170. A block of mass m is connected to another block of mass M by a spring (massless) of spring constant k. The block are kept on a smooth horizontal plane. Initially the blocks are at rest and the spring is unstretched. Then a constant force F starts acting on the block of mass M to pull it. Find the force of the block of mass m.

$\frac{MF}{ \big(m + M\big) }$

$\frac{mF}{M}$

$\frac{ \big(M + m\big)F }{m}$

$\frac{mF}{ \big(m + M\big) }$

171. A block of mass m is placed on a surface with a vertical cross section given by $y = \frac{ x^{3} }{6}$ . If the coefficient of friction is 0.5, the maximum height above the ground at which the block can be placed without slipping is:

$\frac{1}{6}m$

$\frac{2}{3}m$

$\frac{1}{3}m$

$\frac{1}{2}m$

172. A ball of mass 10 g moving perpendicular to the plane of the wall strikes it and rebounds in the same line with the same velocity. If the impulse experienced by the wall is 0.54 Ns, the velocity of the ball is

27 ms^–1

3.7 ms^–1

54 ms^–1

37 ms^–1

173. A block is kept on a inclined plane of inclination $\theta$ of length $\ell$ . The velocity of particle at the bottom of inclined is (the coefficient of friction is $\mu$ )

$\begin{bmatrix}2g\ell \big( \mu cos \theta - sin \theta \big) \end{bmatrix}^{1/2}$

$\sqrt{2g\ell \big( sin \theta - \mu cos \theta \big)}$

$\sqrt{2g\ell \big( sin \theta + \mu cos \theta \big)}$

$\sqrt{2g\ell \big( cos \theta + \mu sin \theta \big)}$

174. A 100 g iron ball having velocity 10 m/s collides with a wall at an angle 30° and rebounds with the same angle. If the period of contact between the ball and wall is 0.1 second, then the force experienced by the wall is

10 N

100 N

1.0 N

0.1 N

175. A bullet is fired from a gun. The force on the bullet is given by F = 600 – 2 × 10⁵ t where, F is in newton and t in second. The force on the bullet becomes zero as soon as it leaves the barrel. What is the average impulse imparted to the bullet?

1.8 N-s

zero

9 N-s

0.9 N-s

176. Two stones of masses m and 2 m are whirled in horizontal circles, the heavier one in radius $\frac{r}{2}$ and the lighter one in radius r. The tangential speed of lighter stone is n times that of the value of heavier stone when they experience same centripetal forces. The value of n is :

177. A 0.1 kg block suspended from a massless string is moved first vertically up with an acceleration of 5ms^-2 and then moved vertically down with an acceleration of 5ms^-2. If $T_{1}$ and $T_{2}$ are the respective tensions in the two cases, then

$T_{2}$ > $T_{1}$

$T_{1}$ - $T_{2}$ = 1 N, if g = 10ms^-2

$T_{1}$ - $T_{2}$ = 1kg f

$T_{1}$ - $T_{2}$ = 9.8 N, if g = 9.8 ms^-2

178. Three forces start acting simultaneously on a particle moving with velocity, $\vec{v}$ . These forces are represented in magnitude and direction by the three sides of a triangle ABC. The particle will now move with velocity

less than $\vec{v}$

greater than $\vec{v}$

|v| in the direction of the largest force BC

$\vec{v}$ , remaining unchanged

179. If in a stationary lift, a man is standing with a bucket full of water, having a hole at its bottom. The rate of flow of water through this hole is $R_{0}$ . If the lift starts to move up and down with same acceleration and then the rates of flow of water are $R_{u}$ and $R_{d}$ , then

$R_{0}$ > $R_{u}$ > $R_{d}$

$R_{u}$ > $R_{0}$ > $R_{d}$

$R_{d}$ > $R_{0}$ > $R_{u}$

$R_{u}$ > $R_{d}$ > $R_{0}$

180. A stationary body of mass 3 kg explodes into three equal pieces. Two of the pieces fly off in two mutually perpendicular directions, one with a velocity of $3\hat{i}$ ms^-1 and the other with a velocity of $4\hat{j}$ ms^-1 . If the explosion occurs in 10^-4 s, the average force acting on the third piece in newton is

$\big(3\hat{i} + 4\hat{j} \big) \times 10^{-4}$

$\big(3\hat{i} - 4\hat{j} \big) \times 10^{-4}$

$\big(3\hat{i} - 4\hat{j} \big) \times 10^{4}$

$-\big(3\hat{i} + 4\hat{j} \big) \times 10^{4}$

181. A spring of spring constant 5 × 10³ N/m is stretched initially by 5cm from the unstretched position. Then the work required to stretch it further by another 5 cm is

12.50 Nm

18.75 Nm

25.00 Nm

6.25 Nm

182. A particle of mass 10 g moves along a circle of radius 6.4 cm with a constant tangential acceleration. What is the magnitude of this acceleration if the kinetic energy of the particle becomes equal to 8 × 10^-4 J by the end of the second revolution after the beginning of the motion ?

0.1 m/s²

0.15 m/s²

0.18 m/s²

0.2 m/s²

183. A body is moved along a straight line by a machine delivering a constant power. The distance moved by the body in time ‘t’ is proportional to

t^3/4

t^3/2

t^1/4

t^1/2

184. A ball is thrown vertically downwards from a height of 20 m with an initial velocity $v_{0}$ . It collides with the ground and loses 50% of its energy in collision and rebounds to the same height. The initial velocity $v_{0}$ is : (Take g = 10 ms^-2)

20 ms^-1

28 ms^-1

10 ms^-1

14 ms^-1

185. A cord is used to lower vertically a block of mass M, a distance d at a constant downward acceleration of g/4. The work done by the cord on the block is

Mg $\frac{d}{4}$

3Mg $\frac{d}{4}$

-3Mg $\frac{d}{4}$

Mg d

186. A rubber ball is dropped from a height of 5m on a plane, where the acceleration due to gravity is not shown. On bouncing it rises to 1.8 m. The ball loses its velocity on bouncing by a factor of

$\frac{16}{25}$

$\frac{2}{5}$

$\frac{3}{5}$

$\frac{9}{25}$

187. A ball of mass m moving with a constant velocity strikes against a ball of same mass at rest. If e = coefficient of restitution, then what will be the ratio of velocity of two balls after collision?

$\frac{1-e}{1+e}$

$\frac{e-1}{e+1}$

$\frac{1+e}{1-e}$

$\frac{2+e}{e-1}$

188. A particle of mass m is driven by a machine that delivers a constant power of k watts. If the particle starts from rest the force on the particle at time t is

$\sqrt{mk}$ t^-1/2

$\sqrt{2mk}$ t^-1/2

$\frac{1}{2}\sqrt{mk}$ t^-1/2

$\sqrt{ \frac{mk}{2}}$ t^-1/2

189. A body of mass 2 kg moving under a force has relation between displacement x and time t as x = $\frac{ t^{3} }{3}$ where x is in metre and t is in sec. The work done by the body in first two second will be

1.6 joule

16 joule

160 joule

1600 joule

190. A sphere of mass 8m collides elastically (in one dimension) with a block of mass 2m. If the initial energy of sphere is E. What is the final energy of sphere?

0.8 E

0.36 E

0.08 E

0.64 E

191. Two similar springs P and Q have spring constants $K_{P}$ and $K_{Q}$ , such that $K_{P}$ > $K_{Q}$ . They are stretched, first by the same amount (case a,) then by the same force (case b). The work done by the springs $W_{P}$ and $W_{Q}$ are related as, in case (a) and case (b), respectively

$W_{P} = W_{Q} ; W_{P} = W_{Q}$

$W_{P} > W_{Q} ; W_{Q} > W_{P}$

$W_{P} < W_{Q} ; W_{Q} < W_{P}$

$W_{P} = W_{Q} ; W_{P} > W_{Q}$

192. In the figure, the variation of potential energy of a particle of mass m = 2 kg is represented w.r.t. its x-coordinate. The particle moves under the effect of this conservative force along the x-axis.

If the particle is released at the origin then

it will move towards positive x-axis

it will move towards negative x-axis

it will remain stationary at the origin

its subsequent motion cannot be decided due to lack of information

193. The potential energy of a certain spring when stretched through distance S is 10 joule. The amount of work done (in joule) that must be done on this spring to stretch it through an additional distance s, will be

194. A force applied by an engine of a train of mass 2.05×10⁶ kg changes its velocity from 5m/s to 25 m/s in 5 minutes. The power of the engine is

1.025 MW

2.05 MW

5 MW

6 MW

195. The relationship between the force F and position x of a body is as shown in figure. The work done in displacing the body form x = 1 m to x = 5 m will be

30 J

15 J

25 J

20 J

196. A body is allowed to fall freely under gravity from a height of 10m. If it looses 25% of its energy due to impact with the ground, then the maximum height it rises after one impact is

2.5m

5.0m

7.5m

8.2m

197. A block C of mass m is moving with velocity $v_{0}$ and collides elastically with block A of mass m and connected to another block B of mass 2m through spring constant k. What is k if $x_{0}$ is compression of spring when velocity of A and B is same?

$\frac{mv_{0}^{2} }{ x_{0}^{2} }$

$\frac{mv_{0}^{2} }{ 2x_{0}^{2} }$

$\frac{3 mv_{0}^{2} }{2 x_{0}^{2} }$

$\frac{2 mv_{0}^{2} }{3 x_{0}^{2} }$

198. Two springs of force constants 300 N/m (Spring A) and 400 N/m (Spring B) are joined together in series. The combination is compressed by 8.75 cm. The ratio of energy stored in A and B is $\frac{ E_{A} }{ E_{B} }$ . Then $\frac{ E_{A} }{ E_{B} }$ is equal to :

$\frac{4}{3}$

$\frac{16}{9}$

$\frac{3}{4}$

$\frac{9}{16}$

199. A body of mass 1 kg begins to move under the action of a time dependent force $\vec{F} = \big(2t \hat{i} + 3t^{2} \hat{j} \big)N$ , where $\hat{i}$ and $\hat{j}$ are unit vectors alogn x and y axis. What power will be developed by the force at the time t?

$\big(2t^{2} + 3t^{3} \big)W$

$\big(2t^{2} + 4t^{4} \big)W$

$\big(2t^{3} + 3t^{4} \big)W$

$\big(2t^{3} + 3t^{5} \big)W$

200. A bullet of mass 20 g and moving with 600 m/s collides with a block of mass 4 kg hanging with the string. What is the velocity of bullet when it comes out of block, if block rises to height 0.2 m after collision?

200 m/s

150 m/s

400 m/s

300 m/s

201. A body of mass m kg is ascending on a smooth inclined plane of inclination $\theta \big( {sin \theta = \frac{1}{x} } \big)$ with constant acceleration of a m/s². The final velocity of the body is v m/s. The work done by the body during this motion is (Initial velocity of the body = 0)

$\frac{1}{2}mv^{2} \big(g + xa\big)$

$\frac{mv^{2}}{2} \big( \frac{g}{2} + a \big)$

$\frac{2mv^{2}x}{a} \big( a +gx \big)$

$\frac{mv^{2}}{2ax} \big(g + xa \big)$

202. A glass marble dropped from a certain height above the horizontal surface reaches the surface in time t and then continues to bounce up and down. The time in which the marble finally comes to rest is

eⁿt

e²t

$t \begin{bmatrix} \frac{1 + e}{1 - e} \end{bmatrix}$

$t \begin{bmatrix} \frac{1 - e}{1 + e} \end{bmatrix}$

203. The potential energy of a 1 kg particle free to move along the x-axis is given by $V \big(x\big) = \big(\frac{ x^{4} }{4} - \frac{ x^{2} }{2}\big)J$ . The total mechanical energy of the particle is 2 J. Then, the maximum speed (in m/s) is

$\frac{3}{ \sqrt{2} }$

$\sqrt{2}$

$\frac{1}{ \sqrt{2} }$

204. Water falls from a height of 60 m at the rate of 15 kg/s to operate a turbine. The losses due to frictional force are 10% of energy. How much power is generated by the turbine?( g = 10 m/s²)

8.1 kW

10.2 kW

12.3 kW

7.0 kW

205. A car of mass m starts from rest and accelerates so that the instantaneous power delivered to the car has a constant magnitude $P_{o}$ . The instantaneous velocity of this car is proportional to :

t² $P_{o}$

t^1/2

t^-1/2

$\frac{t}{ \sqrt{m} }$

206. When a 1.0kg mass hangs attached to a spring of length 50 cm, the spring stretches by 2 cm. The mass is pulled down until the length of the spring becomes 60 cm. What is the amount of elastic energy stored in the spring in this condition. if g = 10 m/s².

1.5 joule

2.0 joule

2.5 joule

3.0 joule

207. A block of mass m rests on a rough horizontal surface (Coefficient of friction is μ). When a bullet of mass m/2 strikes horizontally, and get embedded in it, the block moves a distance d before coming to rest. The initial velocity of the bullet is k $\sqrt{2\mu gd}$ , then the value of k is

208. A force acts on a 30 gm particle in such a way that the position of the particle as a function of time is given by x = 3t – 4t² + t³, where x is in metres and t is in seconds. The work done during the first 4 seconds is

576 mJ

450 mJ

490 mJ

530 mJ

209. A particle of mass $m_{1}$ moving with velocity v strikes with a mass $m_{2}$ at rest, then the condition for maximum transfer of kinetic energy is

$m_{1}$ >> $m_{2}$

$m_{2}$ >> $m_{2}$

$m_{1}$ = $m_{2}$

$m_{1}$ >> $2m_{2}$

210. A mass m is moving with velocity v collides inelastically with a bob of simple pendulum of mass m and gets embedded into it. The total height to which the masses will rise after collision is

$\frac{ v^{2} }{8g}$

$\frac{ v^{2} }{4g}$

$\frac{ v^{2} }{2g}$

$\frac{ 2v^{2} }{g}$

211. A 10 H.P. motor pumps out water from a well of depth 20 m and fills a water tank of volume 22380 litres at a height of 10 m from the ground. The running time of the motor to fill the empty water tank is (g = 10ms^-2)

5 minutes

10 minutes

15 minutes

20 minutes

212. A particle of mass $m_{1}$ is moving with a velocity $v_{1}$ and another particle of mass $m_{2}$ is moving with a velocity $v_{2}$ . Both of them have the same momentum but their different kinetic energies are $E_{1}$ and $E_{2}$ respectively. If $m_{1}$ > $m_{2}$ then

$E_{1}$ = $E_{2}$

$E_{1}$ < $E_{2}$

$\frac{E_{1}}{E_{2}} = \frac{m_{1}}{m_{2}}$

$E_{1}$ > $E_{2}$

213. A block of mass 10 kg, moving in x direction with a constant speed of 10 ms^-1, is subject to a retarding force F = 0.1 × J m during its travel from x = 20 m to 30 m. Its final KE will be :

450 J

275 J

250 J

475 J

214. Identify the false statement from the following

Work-energy theorem is not independent of Newton's second law.

Work-energy theorem holds in all inertial frames.

Work done by friction over a closed path is zero.

No potential energy can be associated with friction.

215. A one-ton car moves with a constant velocity of 15 ms^–1 on a rough horizontal road. The total resistance to the motion of the car is 12% of the weight of the car. The power required to keep the car moving with the same constant velocity of 15ms^–1 is [Take g = 10 ms^–2]

9 kW

18 kW

24 kW

36 kW

216. A ball is released from the top of a tower. The ratio of work done by force of gravity in first, second and third second of the motion of the ball is

1 : 2 : 3

1 : 4 : 9

1 : 3 : 5

1 : 5 : 3

217. Two spheres A and B of masses $m_{1}$ and $m_{2}$ respectively collide. A is at rest initially and B is moving with velocity v along x-axis. After collision B has a velocity $\frac{v}{2}$ in a direction perpendicular to the original direction. The mass A moves after collision in the direction.

Same as that of B

Opposite to that of B

$\theta = tan^{-1}$ (1/2) to the x-axis

$\theta = tan^{-1}$ (–1/2) to the x-axis

218. A 2 kg block slides on a horizontal floor with a speed of 4m/s. It strikes a uncompressed spring, and compresses it till the block is motionless. The kinetic friction force is 15N and spring constant is 10,000 N/m. The spring compresses by

8.5 cm

5.5 cm

2.5 cm

11.0 cm

219. An engine pumps water through a hose pipe. Water passes through the pipe and leaves it with a velocity of 2 m/s. The mass per unit length of water in the pipe is 100 kg/m. What is the power of the engine?

400 W

200 W

100 W

800 W

220. A uniform chain of length 2 m is kept on a table such that a length of 60 cm hangs freely from the edge of the table. The total mass of the chain is 4 kg. What is the work done in pulling the entire chain on the table ?

12 J

3.6 J

7.2 J

1200 J

221. A mass ‘m’ moves with a velocity ‘v’ and collides inelastically with another identical mass. After collision the l^st mass moves with velocity $\frac{v}{ \sqrt{3}}$ in a direction perpendicular to the initial direction of motion. Find the speed of the 2^nd mass after collision.

$\sqrt{3}v$

$\frac{v}{\sqrt{3}}$

$\frac{2}{\sqrt{3}}$ v

222. A spherical ball of mass 20 kg is stationary at the top of a hill of height 100 m. It rolls down a smooth surface to the ground, then climbs up another hill of height 30 m and finally rolls down to a horizontal base at a height of 20 m above the ground. The velocity attained by the ball is

20 m/s

40 m/s

$10\sqrt{30}$ m/s

10 m/s

223. A block of mass M is kept on a platform which is accelerated upward with a constant acceleration 'a' during the time interval T. The work done by normal reaction between the block and platform is

$-\frac{ MgaT^{2} }{2}$

$\frac{1}{2}M \big(g + a\big) aT^{2}$

$\frac{1}{2}Ma^{2}T$

Zero

224. A spring lies along an x axis attached to a wall at one end and a block at the other end. The block rests on a frictionless surface at x = 0. A force of constant magnitude F is applied to the block that begins to compress the spring, until the block comes to a maximum displacement $x_{max}$ .

During the displacement, which of the curves shown in the graph best represents the kinetic energy of the block ?

225. The K.E. acquired by a mass m in travelling a certain distance d, starting form rest, under the action of a constant force is directly proportional to

$\sqrt{m}$

$\frac{1}{\sqrt{m}}$

independent of m

226. A vertical spring with force constant k is fixed on a table. A ball of mass m at a height h above the free upper end of the spring falls vertically on the spring so that the spring is compressed by a distance d. The net work done in the process is

mg(h + d)- $\frac{1}{2} kd^{2}$

mg(h - d)- $\frac{1}{2} kd^{2}$

mg(h - d)+ $\frac{1}{2} kd^{2}$

mg(h + d)+ $\frac{1}{2} kd^{2}$

227. From a solid sphere of mass M and radius R, a cube of maximum possible volume is cut. Moment of inertia of cube about an axis passing through its center and perpendicular to one of its faces is :

$\frac{4MR^{2}}{9\sqrt{3} \pi }$

$\frac{4MR^{2}}{3\sqrt{3} \pi }$

$\frac{MR^{2}}{32\sqrt{2} \pi }$

$\frac{MR^{2}}{16\sqrt{2} \pi }$

228. A hollow sphere is held suspended. Sand is now poured into it in stages. The centre of mass of the sphere with the sand

rises continuously

remains unchanged in the process

first rises and then falls to the original position

first falls and then rises to the original position

229. A body A of mass M while falling vertically downwards under gravity breaks into two parts; a body B of mass $\frac{1}{3} M$ and a body C of mass $\frac{2}{3} M$ . The centre of mass of bodies B and C taken together shifts compared to that of body A towards

does not shift

depends on height of breaking

body B

body C

230. From a uniform wire, two circular loops are made (i) P of radius r and (ii) Q of radius nr. If the moment of inertia of Q about an axis passing through its centre and perpendicular to its plane is 8 times that of P about a similar axis, the value of n is (diameter of the wire is very much smaller than r or nr)

231. A billiard ball of mass m and radius r, when hit in a horizontal direction by a cue at a height h above its centre, acquired a linear velocity $v_{0}$ . The angular velocity $\omega_{0}$ acquired by the ball is

$\frac{5v_{0} r^{2}}{2h}$

$\frac{2v_{0} r^{2}}{5h}$

$\frac{2v_{0}h }{5r^{2}}$

$\frac{5v_{0}h }{2r^{2}}$

232. Three bricks each of length L and mass M are arranged as shown from the wall. The distance of the centre of mass of the system from the wall is

L/4

L/2

(3/2)L

(11/12)L

233. Four point masses, each of value m, are placed at the corners of a square ABCD of side $\ell$ . The moment of inertia of this system about an axis passing through A and parallel to BD is

$2m\ell^{2}$

$\sqrt{3}m\ell^{2}$

$3m\ell^{2}$

$m\ell^{2}$

234. A loop of radius r and mass m rotating with an angular velocity $\omega_{0}$ is placed on a rough horizontal surface. The initial velocity of the centre of the hoop is zero.What will be the velocity of the centre of the hoop when it ceases to slip?

$\frac{r\omega_{0}}{4}$

$\frac{r\omega_{0}}{3}$

$\frac{r\omega_{0}}{2}$

$r \omega_{0}$

235. Two masses $m_{1}$ and $m_{2}$ are connected by a massless spring of spring constant k and unstretched length $\ell$ . The masses are placed on a frictionless straight channel, which are consider our x-axis. They are initially at x = 0 and x = $\ell$ respectively. At t = 0, a velocity $v_{0}$ is suddenly imparted to the first particle. At a later time t, the centre of mass of the two masses is at :

$x = \frac{ m_{2} \ell}{m_{1} + m_{2}}$

$x = \frac{ m_{1} \ell}{m_{1} + m_{2}}$ + $\frac{ m_{2} v_{0}t }{ m_{1} + m_{2} }$

$x = \frac{ m_{2} \ell}{m_{1} + m_{1}}$ + $\frac{ m_{2} v_{0}t }{ m_{1} + m_{2} }$

$x = \frac{ m_{2} \ell}{m_{1} + m_{2}}$ + $\frac{ m_{1} v_{0}t }{ m_{1} + m_{2} }$

236. A body of mass 1.5 kg rotating about an axis with angular velocity of 0.3 rad s^–1 has the angular momentum of 1.8 kg m²s^–1. The radius of gyration of the body about an axis is

2 m

1.2 m

0.2 m

1.6 m

237. If $\vec{F}$ is the force acting on a particle having position vector $\vec{r}$ and $\vec{\tau}$ be the torque of this force about the origin, then:

$\vec{r}$ . $\vec{\tau}$ > 0 and $\vec{F}$ . $\vec{\tau}$ < 0

$\vec{r}$ . $\vec{\tau}$ = 0 and $\vec{F}$ . $\vec{\tau}$ = 0

$\vec{r}$ . $\vec{\tau}$ = 0 and $\vec{F}$ . $\vec{\tau}$ $\neq$ 0

$\vec{r}$ . $\vec{\tau}$ $\neq$ 0 and $\vec{F}$ . $\vec{\tau}$ = 0

238. A thin uniform rod of length l and mass m is swinging freely about a horizontal axis passing through its end. Its maximum angular speed is $\omega$ . Its centre of mass rises to a maximum height of

$\frac{1}{6} \frac{ l\omega}{g}$

$\frac{1}{2} \frac{ l^{2} \omega^{2} }{g}$

$\frac{1}{6} \frac{ l^{2} \omega^{2} }{g}$

$\frac{1}{3} \frac{ l^{2} \omega^{2} }{g}$

239. A wheel is rolling straight on ground without slipping. If the axis of the wheel has speed v, the instantenous velocity of a point P on the rim, defined by angle $\theta$ , relative to the ground will be

v cos $\big(\frac{1}{2} \theta\big)$

2v cos $\big(\frac{1}{2} \theta\big)$

v(1+ sin $\theta$ )

v(1+ cos $\theta$ )

240. A solid sphere having mass m and radius r rolls down an inclined plane. Then its kinetic energy is

$\frac{5}{7}$ rotational and $\frac{2}{7}$ translational

$\frac{2}{7}$ rotational and $\frac{5}{7}$ translational

$\frac{2}{5}$ rotational and $\frac{3}{5}$ translational

$\frac{1}{2}$ rotational and $\frac{1}{2}$ translational

241. A ring of mass M and radius R is rotating about its axis with angular velocity $\omega$ . Two identical bodies each of mass m are now gently attached at the two ends of a diameter of the ring. Because of this, the kinetic energy loss will be :

$\frac{m\big(M + 2m\big) }{M} \omega^{2} R^{2}$

$\frac{Mm}{ \big(M + m\big) } \omega^{2} R^{2}$

$\frac{Mm}{ \big(M + 2m\big) } \omega^{2} R^{2}$

$\frac{ \big(M + m\big)M }{ \big(M + 2m\big) } \omega^{2} R^{2}$

242. Acertain bicycle can go up a gentle incline with constant speed when the frictional force of ground pushing the rear wheel is $F_{2}$ = 4 N. With what force $F_{1}$ must the chain pull on the sprocket wheel if $R_{1}$ =5 cm and $R_{2}$ = 30 cm?

4 N

24 N

140 N

$\frac{35}{4}N$

243. A wooden cube is placed on a rough horizontal table, a force is applied to the cube. Gradually the force is increased. Whether the cube slides before toppling or topples before sliding is independent of :

the position of point of application of the force

the length of the edge of the cube

mass of the cube

Coefficient of friction between the cube and the table

244. From a circular ring of mass M and radius R, an arc corresponding to a 90° sector is removed. The moment of inertia of the ramaining part of the ring about an axis passing through the centre of the ring and perpendicular to the plane of the ring is k times MR². Then the value of k is

3/4

7/8

1/4

245. A mass m moves in a circle on a smooth horizontal plane with velocity $v_{0}$ at a radius $R_{0}$ . The mass is attached to string which passes through a smooth hole in the plane as shown. The tension in the string is increased gradually and finally m moves in a circle of radius $\frac{ R_{0}}{2}$ . The final value of the kinetic energy is

$\frac{1}{4}mv_{0}^{2}$

$2mv_{0}^{2}$

$\frac{1}{2}mv_{0}^{2}$

$mv_{0}^{2}$

246. A rod PQ of length L revolves in a horizontal plane about the axis YY´. The angular velocity of the rod is $\omega$ . If A is the area of cross-section of the rod and $\rho$ be its density, its rotational kinetic energy is

$\frac{1}{3}AL^{3} \rho \omega^{2}$

$\frac{1}{2}AL^{3} \rho \omega^{2}$

$\frac{1}{24}AL^{3} \rho \omega^{2}$

$\frac{1}{18}AL^{3} \rho \omega^{2}$

247. A solid sphere of mass 2 kg rolls on a smooth horizontal surface at 10 m/s. It then rolls up a smooth inclined plane of inclination 30° with the horizontal. The height attained by the sphere before it stops is

700 cm

701 cm

7.1 m

70 m

248. A hollow smooth uniform sphere A of mass m rolls without sliding on a smooth horizontal surface. It collides head on elastically with another stationary smooth solid sphere B of the same mass m and same radius. The ratio of kinetic energy of B to that of A just after the collision is

1 : 1

2 : 3

3 : 2

4 : 3

249. Two discs of same thickness but of different radii are made of two different materials such that their masses are same. The densities of the materials are in the ratio of 1 : 3. The moments of inertia of these discs about the respective axes passing through their centres and perpendicular to their planes will be in the ratio of

1 : 3

3 : 1

1 : 9

9 : 1

250. A pulley fixed to the ceiling carries a string with blocks of mass m and 3 m attached to its ends. The masses of string and pulley are negligible. When the system is released, its centre of mass moves with what acceleration ?

– g/4

g/2

– g/2

251. A ring of mass m and radius R has four particles each of mass m attached to the ring as shown in figure. The centre of ring has a speed $v_{0}$ . The kinetic energy of the system is

$mv_{0}^{2}$

$3mv_{0}^{2}$

$5mv_{0}^{2}$

$6mv_{0}^{2}$

252. Consider a uniform square plate of side ‘a’ and mass ‘M’. The moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corners is

$\frac{5}{6} Ma^{2}$

$\frac{1}{12} Ma^{2}$

$\frac{7}{12} Ma^{2}$

$\frac{2}{3} Ma^{2}$

253. A dancer is standing on a stool rotating about the vertical axis passing through its centre. She pulls her arms towards the body reducing her moment of inertia by a factor of n. The new angular speed of turn table is proportional to

n^–1

n⁰

n²

254. A uniform square plate has a small piece Q of an irregular shape removed and glued to the centre of the plate leaving a hole behind. Then the moment of inertia about the z-axis

increases

decreases

remains same

changed in unpredicted manner.

255. A circular turn table has a block of ice placed at its centre. The system rotates with an angular speed $\omega$ about an axis passing through the centre of the table. If the ice melts on its own without any evaporation, the speed of rotation of the system

becomes zero

remains constant at the same value $\omega$

increases to a value greater than $\omega$

decreases to a value less than $\omega$

256. Seven identical coins are rigidly arranged on a flat table in the pattern shown below so that each coin touches it neighbors. Each coin is a thin disc of mass m and radius r. The moment of inertia of the system of seven coins about an axis that passes through point P and perpendicular to the plane of the coin is :

$\frac{55}{2} mr^{2}$

$\frac{127}{2} mr^{2}$

$\frac{111}{2} mr^{2}$

$55mr^{2}$

257. In a two-particle system with particle masses $m_{1}$ and $m_{2}$ , the first particle is pushed towards the centre of mass through a distance d, the distance through which second particle must be moved to keep the centre of mass at the same position is

$\frac{ m_{2}d }{ m_{1} }$

$\frac{ m_{1}d }{ \big(m_{1} + m_{2} \big) }$

$\frac{ m_{1}d }{ m_{2} }$

258. A uniform bar of mass M and length L is horizontally suspended from the ceiling by two vertical light cables as shown. Cable A is connected 1/4th distance from the left end of the bar. Cable B is attached at the far right end of the bar. What is the tension in cable A?

1/4 Mg

1/3 Mg

2/3 Mg

3/4 Mg

259. A couple produces

purely linear motion

purely rotational motion

linear and rotational motion

no motion

260. Point masses 1, 2, 3 and 4 kg are lying at the point (0, 0, 0), (2, 0, 0), (0, 3, 0) and (–2, –2, 0) respectively. The moment of inertia of this system about x-axis will be

43 kgm²

34 kgm²

27 kgm²

72 kgm²

261. A solid sphere of mass M and radius R is pulled horizontally on a sufficiently rough surface as shown in the figure. Choose the correct alternative.

The acceleration of the centre of mass is F/M

The acceleration of the centre of mass is $\ \frac{2}{3} \frac{F}{M}$

The friction force on the sphere acts forward

The magnitude of the friction force is F/3

262. The moment of inertia of a body about a given axis is 1.2 kg m². Initially, the body is at rest. In order to produce a rotational kinetic energy of 1500 joule, an angular acceleration of 25 radian/sec² must be applied about that axis for a duration of

4 sec

2 sec

8 sec

10 sec

263. A gymnast takes turns with her arms and legs stretched. When she pulls her arms and legs in

the angular velocity decreases

the moment of inertia decreases

the angular velocity stays constant

the angular momentum increases

264. An equilateral triangle ABC formed from a uniform wire has two small identical beads initially located at A. The triangle is set rotating about the vertical axis AO. Then the beads are released from rest simultaneously and allowed to slide down, one along AB and the other along AC as shown. Neglecting frictional effects, the quantities that are conserved as the beads slide down, are

angular velocity and total energy (kinetic and potential)

total angular momentum and total energy

angular velocity and moment of inertia about the axis of rotation

total angular momentum and moment of inertia about the axis of rotation

265. The moment of inertia of a uniform semicircular wire of mass m and radius r, about an axis passing through its centre of mass and perpendicular to its plane is $mr^{2} \big(1- \frac{k}{ \pi^{2}} \big)$ . Find the value of k.

266. Initial angular velocity of a circular disc of mass M is $\omega_{1}$ . Then two small spheres of mass m are attached gently to diametrically opposite points on the edge of the disc. What is the final angular velocity of the disc?

$\big( \frac{M + m}{M} \big) \omega_{1}$

$\big( \frac{M + m}{m} \big) \omega_{1}$

$\big( \frac{M}{M + 4m} \big) \omega_{1}$

$\big( \frac{M}{M + 2m} \big) \omega_{1}$

267. Two identical discs of mass m and radius r are arranged as shown in the figure. If $\alpha$ is the angular acceleration of the lower disc and $a_{cm}$ is acceleration of centre of mass of the lower disc, then relation between $a_{cm}$ , $\alpha$ and r is

$a_{cm}$ = $\alpha/r$

$a_{cm}$ = $2\alpha r$

$a_{cm}$ = $\alpha r$

None of these

268. Five masses are placed in a plane as shown in figure. The coordinates of the centre of mass are nearest to

1.2, 1.4

1.3, 1.1

1.1, 1.3

1.0, 1.0

269. Three particles, each of mass m gram, are situated at the vertices of an equilateral triangle ABC of side $\ell$ cm (as shown in the figure). The moment of inertia of the system about a line AX perpendicular to AB and in the plane of ABC, in gram-cm² units will be

$\frac{3}{2}m\ell^{2}$

$\frac{3}{4}m\ell^{2}$

$2m\ell^{2}$

$\frac{5}{4}m\ell^{2}$

270. When a ceiling fan is switched on, it makes 10 rotations in the first 3 seconds. Assuming a uniform angular acceleration, how many rotation it will make in the next 3 seconds?

271. A solid sphere spinning about a horizontal axis with an angular velocity $\omega$ is placed on a horizontal surface. Subsequently it rolls without slipping with an angular velocity of :

$\frac{2\omega}{5}$

$\frac{7\omega}{5}$

$\frac{2\omega}{7}$

$\omega$

272. The radius of a planet is 1/4th of $R_{e}$ and its acc. due to gravity is 2g. What would be the value of escape velocity on the planet, if escape velocity on earth is $v_{e}$ .

$\frac{v_{e}}{ \sqrt{2} }$

$v_{e} \sqrt{2}$

$2v_{e}$

$\frac{v_{e}}{2}$

273. A projectile is fired vertically from the Earth with a velocity $kv_{e}$ where $v_{e}$ is the escape velocity and k is a constant less than unity. The maximum height to which projectile rises, as measured from the centre of Earth, is

$\frac{R}{k}$

$\frac{R}{k-1}$

$\frac{R}{1 - k^{2} }$

$\frac{R}{1 + k^{2} }$

274. A solid sphere of uniform density and radius R applies a gravitational force of attraction equal to $F_{1}$ on a particle placed at A, distance 2R from the centre of the sphere. A spherical cavity of radius R/2 is now made in the sphere as shown in the figure. The sphere with cavity now applies a gravitational force $F_{2}$ on the same particle placed at A. The ratio $F_{2}$ / $F_{1}$ will be

1/2

1/9

275. A geostationary satellite is orbiting the earth at a height of 5R above that surface of the earth, R being the radius of the earth. The time period of another satellite in hours at a height of 2R from the surface of the earth is :

$6\sqrt{2} }$

$\frac{6}{ \sqrt{2} }$

276. A satellite of mass m is orbiting around the earth in a circular orbit with a velocity v. What will be its total energy?

(3/4) mv²

(1/2) mv²

mv²

– (1/2)m v²

277. The gravitational force of attraction between a uniform sphere of mass M and a uniform rod of length l and mass m oriented as shown is

$\frac{GMm}{ \big r(r + l\big) }$

$\frac{GM}{ r^{2} }$

$Mmr^{2}+l$

$\big( r^{2} + l\big)mM$

278. If the gravitational force between two objects were proportional to 1/R (and not as 1/R²) where R is separation between them, then a particle in circular orbit under such a force would have its orbital speed v proportional to

1/R²

R⁰

R¹

1/R

279. A satellite of mass m revolves around the earth of radius R at a height ‘x’ from its surface. If g is the acceleration due to gravity on the surface of the earth, the orbital speed of the satellite is

$\frac{gR^{2}}{R + x}$

$\frac{gR}{R- x}$

$\big(\frac{gR^{2}}{R + x}\big)^{1/2}$

280. A body is projected up with a velocity equal to 3/4th of the escape velocity from the surface of the earth. The height it reaches from the centre of the earth is (Radius of the earth = R)

$\frac{10R}{9}$

$\frac{16R}{7}$

$\frac{9R}{8}$

$\frac{10R}{3}$

281. A Planet is revolving around the sun.

Which of the following is correct option?

The time taken in travelling DAB is less than that for BCD

The time taken in travelling DAB is greater than that for BCD

The time taken in travelling CDA is less than that for ABC

The time taken in travelling CDA is greater than that for ABC

282. The acceleration due to gravity on the planet A is 9 times the acceleration due to gravity on planet B. A man jumps to a height of 2m on the surface of A. What is the height of jump by the same person on the planet B?

$\frac{2}{3}m$

$\frac{2}{9}m$

18 m

6 m

283. If suddenly the gravitational force of attraction between the earth and a satellite revolving around it becomes zero, then the satellite will

continue to move in its orbit with same speed

move tangentially to the original orbit with same speed

become stationary in its orbit

move towards the earth

284. Mass M is divided into two parts xM and (1 – x )M. For a given separation, the value of x for which the gravitational attraction between the two pieces becomes maximum is

$\frac{1}{2}$

$\frac{3}{5}$

285. The potential energy of a satellite, having mass m and rotating at a height of 6.4 × 10⁶ m from the earth surface, is

– mg $R_{e}$

– 0.67 mg $R_{e}$

– 0.5 mg $R_{e}$

– 0.33 mg $R_{e}$

286. If the radius of the earth were to shrink by 1%, with its mass remaining the same, the acceleration due to gravity on the earth’s surface would

decrease by 1%

decrease by 2%

increase by 1%

increase by 2%

287. Suppose the law of gravitational attraction suddenly changes and becomes an inverse cube law i.e. $F \propto \frac{1}{ r^{3} }$ , but still remaining a central force. Then

Kepler’s law of area still holds

Kepler’s law of period still holds

Kepler’s law of area and period still holds

neither the law of area nor the law of period still holds

288. Four equal masses (each of mass M) are placed at the corners of a square of side a. The escape velocity of a body from the centre O of the square is

$4\sqrt{ \frac{2GM}{a} }$

$\sqrt{\frac{8\sqrt{2}GM}{a}}$

$\frac{4GM}{a}$

$\sqrt{\frac{4\sqrt{2}GM}{a}}$

289. If the gravitational force had varied as r^–5/2 instead of r^-2; the potential energy of a particle at a distance ‘r’ from the centre of the earth would be directly proportional to

r^-1

r^-2

r^{-3 / 2}

r^{-5 / 2}

290. A particle of mass ‘m’ is kept at rest at a height 3R from the surface of earth, where ‘R’ is radius of earth and ‘M’ is mass of earth. The minimum speed with which it should be projected, so that it does not return back, is (g is acceleration due to gravity on the surface of earth)

$\big(\frac{GM}{R}\big)^{ \frac{1}{2} }$

$\big(\frac{GM}{2R}\big)^{ \frac{1}{2} }$

$\big(\frac{gR}{4}\big)^{ \frac{1}{2} }$

$\big(\frac{2g}{4}\big)^{ \frac{1}{2} }$

291. The ratio between the values of acceleration due to gravity at a height 1 km above and at a depth of 1 km below the Earth’s surface is (radius of Earth is R)

$\frac{R - 2}{R -1}$

$\frac{R}{R -1}$

$\frac{R - 2}{R}$

292. The weight of an object in the coal mine, sea level and at the top of the mountain, are respectively $w_{1}$ , $w_{2}$ and $w_{3}$ then

$w_{1}$ < $w_{2}$ > $w_{3}$

$w_{1}$ = $w_{2}$ = $w_{3}$

$w_{1}$ < $w_{2}$ < $w_{3}$

$w_{1}$ > $w_{2}$ > $w_{3}$

293. The period of moon's rotation around the earth is nearly 29 days. If moon's mass were 2 fold its present value and all other things remain unchanged, the period of moon's rotation would be nearly

$29\sqrt{2}$ days

$29/ \sqrt{2}$ days

29 × 2 days

29 days

294. The mean radius of earth is R, its angular speed on its own axis is $\omega$ and the acceleration due to gravity at earth's surface is g. What will be the radius of the orbit of a geostationary satellite ?

$\big( R^{2}g / \omega^{2} \big)^{1/3}$

$\big( Rg / \omega^{2} \big)^{1/3}$

$\big( R^{2} \omega^{2} /g \big)^{1/3}$

$\big( R^{2}g / \omega \big)^{1/3}$

295. In order to make the effective acceleration due to gravity equal to zero at the equator, the angular velocity of rotation of the earth about its axis should be (g = 10 ms^-2 and radius of earth is 64000 km)

Zero

$\frac{1}{800}$ rad sec^–1

$\frac{1}{80}$ rad sec^–1

$\frac{1}{8}$ rad sec^–1

296. A body weighs 72 N on the surface of the earth. What is the gravitational force on it due to earth at a height equal to half the radius of the earth from the surface?

32 N

28 N

16 N

72 N

297. A body weighs W newton at the surface of the earth. Its weight at a height equal to half the radius of the earth, will be

$\frac{W}{2}$

$\frac{2W}{3}$

$\frac{4W}{9}$

$\frac{8W}{27}$

298. A shell of mass M and radius R has a point mass m placed at a distance r from its centre. The graph of gravitational potential energy U(r) vs distance r will be

299. The largest and the shortest distance of the earth from the sun are $r_{1}$ and $r_{2}$ . Its distance from the sun when it is at perpendicular to the major-axis of the orbit drawn from the sun

$\big(r_{1} + r_{2}\big) /4$

$\big(r_{1} + r_{2}\big) / \big(r_{1} - r_{2}\big)$

2 $r_{1}r_{2}$ / $\big(r_{1} + r_{2}\big)$

$\big(r_{1} + r_{2}\big) /3$

300. A planet is moving in an elliptical orbit around the sun. If T, V, E and L stand respectively for its kinetic energy, gravitational potential energy, total energy and magnitude of angular momentum about the centre of force, then which of the following is correct ?

T is conserved

V is always positive

E is always negative

L is conserved but direction of vector L changes continuously

301. The earth is assumed to be sphere of radius R. A platform is arranged at a height R from the surface of Earth. The escape velocity of a body from this platform is kv, where v is its escape velocity from the surface of the earth. The value of k is

$\frac{1}{ \sqrt{2}}$

$\frac{1}{3}$

$\frac{1}{2}$

$\sqrt{2}$

302. A solid sphere of mass M and radius R is surrounded by a spherical shell of same mass M and radius 2R as shown. A small particle of mass m is released from rest from a height h [ << R] above the shell. There is a hole in the shell. What time will it enter the hole at A ?

$2\sqrt{\frac{ hR^{2} }{GM}}$

$\sqrt{\frac{ 2hR^{2} }{GM}}$

$\sqrt{\frac{ hR^{2} }{GM}}$

$\sqrt{\frac{ 3hR^{2} }{GM}}$

303. A body starts from rest from a point distance $R_{0}$ from the centre of the earth. The velocity acquired by the body when it reaches the surface of the earth will be (R represents radius of the earth).

$2GM \big( \frac{1}{R} - \frac{1}{ R_{0}} \big)$

$\sqrt{2GM \big( \frac{1}{ R_{0}} - \frac{1}{R} \big)}$

$GM \big( \frac{1}{R} - \frac{1}{ R_{0}} \big)$

$2GM\sqrt{ \big( \frac{1}{R} - \frac{1}{ R_{0}} \big)}$

304. A satellite of mass M is moving in a circle of radius R under a centripetal force given by (–k/R²), where k is a constant. Then

The kinetic energy of the particle is $\frac{k}{12}R$

The total energy of the particle is $\big(-\frac{k}{2R}\big)$

The kinetic energy of the particle is $\big(-\frac{k}{R}\big)$

The potential energy of the particle is $\big(\frac{k}{2R}\big)$

305. The change in the value of ‘g’ at a height ‘h’ above the surface of the earth is the same as at a depth ‘d’ below the surface of earth. When both ‘d’ and ‘h’ are much smaller than the radius of earth, then which one of the following is correct?

$d = \frac{3h}{2}$

$d = \frac{h}{2}$

d = h

d =2 h

306. Two identical geostationary satellites are moving with equal speeds in the same orbit but their sense of rotation brings them on a collision course. The debris will

fall down

move up

begin to move from east to west in the same orbit

begin to move from west to east in the same orbit

307. A diametrical tunnel is dug across the Earth. A ball is dropped into the tunnel from one side. The velocity of the ball when it reaches the centre of the Earth is .... (Given : gravitational potential at the centre of Earth = $-\frac{3}{2} \frac{GM}{R}$ )

$\sqrt{R}$

$\sqrt{gR}$

$\sqrt{2.5gR}$

$\sqrt{7.1gR}$

308. A satellite revolves around the earth of radius R in a circular orbit of radius 3R. The percentage increase in energy required to lift it to an orbit of radius 5R is

10 %

20 %

30 %

40 %

309. A (nonrotating) star collapses onto itself from an initial radius $R_{i}$ with its mass remaining unchanged. Which curve in figure best gives the gravitational acceleration $a_{g}$ on the surface of the star as a function of the radius of the star during the collapse

310. If the earth is treated as a sphere of radius R and mass M; its angular momentum about the axis of its rotation with period T, is

$\frac{ \pi MR^{3}}{T}$

$\frac{ MR^{2}\pi}{T}$

$\frac{ 2\pi MR^{2}}{5T}$

$\frac{ 4\pi MR^{2}}{5T}$

311. A satellite is launched into a circular orbit of radius R around the earth. A second satellite is launched into an orbit of radius 1.01 R. The period of second satellite is larger than the first one by approximately

0.5%

1.0%

1.5%

3.0%

312. A uniform spherical shell gradually shrinks maintaining its shape. The gravitational potential at the centre

increases

decreases

remains constant

cannot say

313. The depth d at which the value of acceleration due to gravity becomes $\frac{1}{n}$ times the value at the surface of the earth, is [R = radius of the earth]

$\frac{R}{n}$

$R\big(\frac{n - 1}{n}\big)$

$\frac{R}{ n^{2}}$

$R\big(\frac{n}{n + 1}\big)$

314. Radius of moon is 1/4 times that of earth and mass is 1/81 times that of earth. The point at which gravitational field due to earth becomes equal and opposite to that of moon, is (Distance between centres of earth and moon is 60R, where R is radius of earth)

5.75 R from centre of moon

16 R from surface of moon

53 R from centre of earth

54 R from centre of earth

315. If earth is supposed to be a sphere of radius R, if $g_{30}$ is value of acceleration due to gravity at lattitude of 30° and g at the equator, the value of g – $g_{30}$ is

$\frac{1}{4} \omega^{2}R$

$\frac{3}{4} \omega^{2}R$

$\omega^{2}R$

$\frac{1}{2} \omega^{2}R$

316. What is the minimum energy required to launch a satellite of mass m from the surface of a planet of mass M and radius R in a circular orbit at an altitude of 2R?

$\frac{5GmM}{6R}$

$\frac{2GmM}{3R}$

$\frac{GmM}{2R}$

317. Two wires A and B are of the same material. Their lengths are in the ratio 1 : 2 and the diameter are in the ratio 2 : 1. If they are pulled by the same force, then increase in length will be in the ratio

2 : 1

1 : 4

1 : 8

8 : 1

318. The load versus elongation graphs for four wires of same length and made of the same material are shown in the figure. The thinnest wire is represented by the line

319. A spring of force constant 800 N/m has an extension of 5 cm. The work done in extending it from 5 cm to 15 cm is

16 J

8 J

32 J

24 J

320. A metal wire of length $L_{1}$ and area of cross-section A is attached to a rigid support. Another metal wire of length $L_{2}$ and of the same cross-sectional area is attached to the free end of the first wire. A body of mass M is then suspended from the free end of the second wire. If $Y_{1}$ and $Y_{2}$ are the Young’s moduli of the wires respectively, the effective force constant of the system of two wires is

$\frac{ \big( Y_{1} Y_{2}\big)A }{ 2\big(Y_{1} L_{2} + Y_{2} L_{1}\big) }$

$\frac{ \big( Y_{1} Y_{2}\big)A }{ \big(L_{1} L_{2}\big) ^{1/2} }$

$\frac{ \big( Y_{1} Y_{2}\big)A }{ \big(Y_{1} L_{2} + Y_{2} L_{1}\big) }$

$\frac{ \big( Y_{1} Y_{2}\big)^{1/2}A }{ \big(L_{2} L_{1}\big) ^{1/2} }$

321. The approximate depth of an ocean is 2700 m. The compressibility of water is 45.4 × 10^–11 Pa^–1 and density of water is 10³ kg/m³.What fractional compression of water will be obtained at the bottom of the ocean ?

1.0 × 10^-2

1.2 × 10^-2

1.4 × 10^-2

0.8 × 10^-2

322. The Young's modulus of steel is twice that of brass. Two wires of same length and of same area of cross section, one of steel and another of brass are suspended from the same roof. If we want the lower ends of the wires to be at the same level, then the weights added to the steel and brass wires must be in the ratio of :

2 : 1

4 : 1

1 : 1

1 : 2

323. Choose the wrong statement.

The bulk modulus for solids is much larger than for liquids.

Gases are least compressible.

The incompressibility of the solids is due to the tight coupling between neighbouring atoms.

The reciprocal of the bulk modulus is called compressibility.

324. A copper wire of length 1.0 m and a steel wire of length 0.5 m having equal cross-sectional areas are joined end to end. The composite wire is stretched by a certain load which stretches the copper wire by 1 mm. If the Young’s modulii of copper and steel are respectively 1.0 × 10¹¹ Nm^-2 and 2.0 × 10¹¹ Nm^-2, the total extension of the composite wire is :

1.75 mm

2.0 mm

1.50 mm

1.25 mm

325. A cube at temperature 0ºC is compressed equally from all sides by an external pressure P. By what amount should its temperature be raised to bring it back to the size it had before the external pressure was applied. The bulk modulus of the material of the cube is B and the coefficient of linear expansion is a.

P/B $\alpha$

P/3 B $\alpha$

$3\pi \alpha/B$

3 B/P

326. The diagram below shows the change in the length X of a thin uniform wire caused by the application of stress F at two different temperatures $T_{1}$ and $T_{2}$ . The variation shown suggests that

$T_{1}$ > $T_{2}$

$T_{1}$ < $T_{2}$

$T_{2}$ > $T_{1}$

$T_{1}$ $\geq$ $T_{2}$

327. If the ratio of lengths, radii and Young's moduli of steel and brass wires in the figure are a, b and c respectively, then the corresponding ratio of increase in their lengths is :

$\frac{3c}{2ab^{2}}$

$\frac{2a^{2}c}{b}$

$\frac{3a}{2b^{2}c}$

$\frac{2ac}{ b^{2}}$

328. The Young’s modulus of brass and steel are respectively 10¹⁰ N/m². and 2 × 10¹⁰ N/m². A brass wire and a steel wire of the same length are extended by 1 mm under the same force, the radii of brass and steel wires are $R_{B}$ and $R_{S}$ respectively. Then

$R_{S}$ = $\sqrt{2}$ $R_{B}$

$R_{S}$ = $R_{B}$ / $\sqrt{2}$

$R_{S}$ = 4 $R_{B}$

$R_{S}$ = $R_{B}$ /4

329. Steel ruptures when a shear of 3.5 × 10⁸ N m^-2 is applied. The force needed to punch a 1 cm diameter hole in a steel sheet 0.3 cm thick is nearly:

1.4 × 10⁴ N

2.7 × 10⁴ N

3.3 × 10⁴ N

1.1 × 10⁴ N

330. A ball falling in a lake of depth 400 m has a decrease of 0.2% in its volume at the bottom. The bulk modulus of the material of the ball is (in N m^-2)

9.8 × 10⁹

9.8 × 10¹⁰

1.96 × 10¹⁰

1.96 × 10⁹

331. A circular tube of mean radius 8 cm and thickness 0.04 cm is melted up and recast into a solid rod of the same length. The ratio of the torsional rigidities of the circular tube and the solid rod is

$\frac{ \big(8.02\big)^{4} - \big(7.98\big)^{4} }{\big(0.8\big)^{4} }$

$\frac{ \big(8.02\big)^{2} - \big(7.98\big)^{2} }{\big(0.8\big)^{2} }$

$\frac{\big(0.8\big)^{2}}{\big(8.02\big)^{4} - \big(7.98\big)^{4} }}$

$\frac{\big(0.8\big)^{2}}{\big(8.02\big)^{3} - \big(7.98\big)^{2} }}$

332. Two wires are made of the same material and have the same volume. However wire 1 has cross-sectional area A and wire 2 has cross-sectional area 3A. If the length of wire 1 increases by $\Delta x$ on applying force F, how much force is needed to stretch wire 2 by the same amount?

4 F

6 F

9 F

333. In materials like aluminium and copper, the correct order of magnitude of various elastic modului is:

Young’s modulus < shear modulus < bulk modulus.

Bulk modulus < shear modulus < Young’s modulus

Shear modulus < Young’s modulus < bulk modulus.

Bulk modulus < Young’s modulus < shear modulus.

334. What per cent of length of wire increases by applying a stress of 1 kg weight/mm² on it?

(Y = 1 × 10¹¹ N/m² and 1 kg weight = 9.8 newton)

0.0067%

0.0098%

0.0088%

0.0078%

335. An elastic string of unstretched length L and force constant k is stretched by a small length x. It is further stretched by another small length y. The work done in the second stretching is :

$\frac{1}{2} ky^{2}$

$\frac{1}{2} k\big(x^{2} + y^{2}\big) }$

$\frac{1}{2} k\big(x + y \big)^{2} }$

$\frac{1}{2} ky\big(2x + y \big)$

336. Two, spring P and $\varrho$ of force constants kp and $k \varrho \big(k \varrho = \frac{k_{p}}{2}\big)$ are stretched by applying forces of equal magnitude. If the energy stored in $\varrho$ is E, then the energy stored in P is

E/2

E/4

337. The pressure that has to be applied to the ends of a steel wire of length 10 cm to keep its length constant when its temperature is raised by 100ºC is: (For steel Young’s modulus is 2 × 10¹¹Nm^-2 and coefficient of thermal expansion is 1.1 × 10^-5K^-1 )

2.2 × 10⁸ Pa

2.2 × 10⁹ Pa

2.2 × 10⁷ Pa

2.2 × 10⁶ Pa

338. A steel ring of radius r and cross sectional area A is fitted onto a wooden disc of radius R (R > r). If the Young’s modulus of steel is Y, then the force with which the steel ring is expanded is

A Y (R/r)

A Y (R – r)/r

(Y/A)[(R – r)/r]

Y r/A R

339. Two wires A and B of same material and of equal length with the radii in the ratio 1 : 2 are subjected to identical loads. If the length of A increases by 8 mm, then the increase in length of B is

2 mm

4 mm

8 mm

16 mm

340. A material has poisson’s ratio 0.50. If a uniform rod of it suffers a longitudinal strain of 2 × 10^-3, then the percentage change in volume is

0.6

0.4

0.2

Zero

341. The upper end of a wire of diameter 12mm and length 1m is clamped and its other end is twisted through an angle of 30°. The angle of shear is

18°

0.18°

36°

0.36°

342. The pressure on an object of bulk modulus B undergoing hydraulic compression due to a stress exerted by surrounding fluid having volume strain $\big(\frac{ \Delta V}{V}\big)^{2}$ is

$B^{2}\big(\frac{ \Delta V}{V}\big)$

$B\big(\frac{ \Delta V}{V}\big)^{2}$

$\frac{1}{B} \big(\frac{ \Delta V}{V}\big)$

$B\big(\frac{ \Delta V}{V}\big)$

343. A structural steel rod has a radius of 10 mm and length of 1.0 m. A 100 kN force stretches it along its length. Young’s modulus of structural steel is 2 × 10¹¹ Nm^-2. The percentage strain is about

0.16%

0.32%

0.08%

0.24%

344. A beam of metal supported at the two edges is loaded at the centre. The depression at the centre is proportional to

$Y^{2}$

1/Y

1/ $Y^{2}$

345. When a 4 kg mass is hung vertically on a light spring that obeys Hooke’s law, the spring stretches by 2 cms. The work required to be done by an external agent in stretching this spring by 5 cms will be (g = 9.8 m/sec²)

4.900 joule

2.450 joule

0.495 joule

0.245 joule

346. The length of a metal is $\ell_{1}$ when the tension in it is $T_{1}$ and is $\ell_{2}$ when the tension is $T_{2}$ . The original length of the wire is

$\frac{\ell_{1} + \ell_{2}}{2}$

$\frac{ \ell_{1} T_{2} + \ell_{2} T_{1} }{ T_{1} + T_{2} }$

$\frac{ \ell_{1} T_{2} - \ell_{2} T_{1} }{ T_{2} - T_{1} }$

$\sqrt{T_{1} T_{2} \ell_{1} \ell_{2}}$

347. For the same cross-sectional area and for a given load, the ratio of depressions for the beam of a square cross-section and circular cross-section is

$3 : \pi$

$\pi : 3$

$1 : \pi$

$\pi : 1$

348. The bulk moduli of ethanol, mercury and water are given as 0.9, 25 and 2.2 respectively in units of 10⁹ Nm^-2. For a given value of pressure, the fractional compression in volume is $\frac{\Delta V}{V}$ . Which of the following statements about $\frac{\Delta V}{V}$ for these three liquids is correct ?

Ethanol > Water > Mercury

Water > Ethanol > Mercury

Mercury > Ethanol > Water

Ethanol > Mercury > Water

349. The graph given is a stress-strain curve for

elastic objects

plastics

elastomers

None of these

350. A metal rod of Young's modulus 2 × 10¹⁰ N m^-2 undergoes an elastic strain of 0.06%. The energy per unit volume stored in J m^-3 is

3600

7200

10800

14400

351. Two wires of the same material and same length but diameters in the ratio 1 : 2 are stretched by the same force. The potential energy per unit volume of the two wires will be in the ratio

1 : 2

4 : 1

2 : 1

16 : 1

352. The length of an elastic string is a metre when the longitudinal tension is 4 N and b metre when the longitudinal tension is 5 N. The length of the string in metre when the longitudinal tension is 9 N is

a – b

5b – 4a

$2b-\frac{1}{4}a$

4a – 3b

353. A force of 10³ newton, stretches the length of a hanging wire by 1 millimetre. The force required to stretch a wire of same material and length but having four times the diameter by 1 millimetre is

4 × 10³ N

16 × 10³ N

$\frac{1}{4}\times 10^{3}N$

$\frac{1}{16}\times 10^{3}N$

354. A steel wire of length l and cross sectional area A is stretched by 1 cm under a given load. When the same load is applied to another steel wire of double its length and half of its cross section area, the amount of stretching (extension) is

0.5 cm

2 cm

4 cm

1.5 cm

355. The adjacent graph shows the extension ( $\Delta l$ ) of a wire of length 1 m suspended from the top of a roof at one end with a load W connected to the other end. If the cross-sectional area of the wire is 10^-6 m², calculate the Young’s modulus of the material of the wire :

2 × 10¹¹ N/m²

2 × 10^-11 N/m²

3 × 10^-12 N/m²

2 × 10^-13 N/m²

356. If a rubber ball is taken at the depth of 200 m in a pool, its volume decreases by 0.1%. If the density of the water is 1 × 10³ kg/m³ and g = 10m/s², then the volume elasticity in N/m² will be

10⁸

2 × 10⁸

10⁹

2 × 10⁹

357. A ball is falling in a lake of depth 200 m creates a decrease 0.1 % in its volume at the bottom. The bulk modulus of the material of the ball will be

19.6 × 10^-8 N/m²

19.6 × 10¹⁰ N/m²

19.6 × 10^-10 N/m²

19.6 × 10⁸ N/m²

358. The diagram shows a force extension graph for a rubber band. Consider the following statements :

I. It will be easier to compress this rubber than expand it
II. Rubber does not return to its original length after it is stretched
III. The rubber band will get heated if it is stretched and released

Which of these can be deduced from the graph:

III only

II and III

I and III

I only

359. The Poisson’s ratio of a material is 0.5. If a force is applied to a wire of this material, there is a decrease in the cross-sectional area by 4%. The percentage increase in the length is:

2.5%

360. Copper of fixed volume ‘V; is drawn into wire of length ‘l’. When this wire is subjected to a constant force ‘F’, the extension produced in the wire is ‘ $\Delta l$ ’. Which of the following graphs is a straight line?

$\Delta l$ versus $\frac{1}{l}$

$\Delta l$ versus $l^{2}$

$\Delta l$ versus $\frac{1}{ l^{2}}$

$\Delta l$ versus $l$

361. When a 4 kg mass is hung vertically on a light spring that obeys Hooke’s law, the spring stretches by 2 cms. The work required to be done by an external agent in stretching this spring by 5 cm will be (g = 9.8 m/sec²)

4.900 joule

2.450 joule

0.495 joule

0.245 joule

362. The density of water at the surface of ocean is r. If the bulk modulus of water is B, what is the density of ocean water at a depth where the pressure is n $P_{0}$ , where $P_{0}$ is the atmospheric pressure ?

$\frac{ \rho B}{B - \big(n -1\big) P_{0}}$

$\frac{ \rho B}{B + \big(n -1\big) P_{0}}$

$\frac{ \rho B}{B - nP_{0}}$

$\frac{ \rho B}{B + nP_{0}}$

363. A ball of radius r and density $\rho$ falls freely under gravity through a distance h before entering water. Velocity of ball does not change even on entering water. If viscosity of water is $\eta$ the value of h is given by

$\frac{2}{9}r^{2} \big( \frac{1 - \rho}{\eta} \big)g$

$\frac{2}{81}r^{2} \big( \frac{\rho - 1 }{\eta} \big)g$

$\frac{2}{81}r^{4} \big( \frac{\rho - 1}{\eta} \big)^{2}g$

$\frac{2}{9}r^{4} \big( \frac{\rho - 1}{\eta} \big)^{2}g$

364. Two parallel glass plates are dipped partly in the liquid of density 'd' keeping them vertical . If the distance between the plates is 'x', surface tension for liquids is T and angle of contact is $\theta$ , then rise of liquid between the plates due to capillary will be

$\frac{T cos\theta}{xd}$

$\frac{2T cos\theta}{xdg}$

$\frac{2T} {xdg cos\theta}$

$\frac{T cos\theta} {xdg}$

365. A liquid is allowed to flow into a tube of truncated cone shape. Identify the correct statement from the following

The speed is high at the wider end and high at the narrow end

The speed is low at the wider end and high at the narrow end

The speed is same at both ends in a streamline flow

The liquid flows with uniform velocity in the tube

366. A wide vessel with a small hole at the bottom is filled with water (density $\rho_{1}$ , height $h_{1}$ ) and kerosene (density $\rho_{2}$ , height $h_{2}$ ). Neglecting viscosity effects, the speed with which water flows out is :

$\begin{bmatrix}2g \big(h_{1} + h_{2}\big) \end{bmatrix}^{1/2}$

$\begin{bmatrix}2g \big(h_{1}\rho_{1} + h_{2}\rho_{2}\big) \end{bmatrix}^{1/2}$

$\begin{bmatrix}2g \big(h_{1} + h_{2} \big( \rho_{2}/\rho_{1} \big) \big) \end{bmatrix}^{1/2}$

$\begin{bmatrix}2g \big(h_{1} + h_{2} \big( \rho_{1}/\rho_{2} \big) \big) \end{bmatrix}^{1/2}$

367. A capillary tube of radius r is immersed vertically in a liquid such that liquid rises in it to height h (less than the length of the tube). Mass of liquid in the capillary tube is m. If radius of the capillary tube is increased by 50%, then mass of liquid that will rise in the tube, is

$\frac{2}{3}m$

$\frac{4}{9}m$

$\frac{3}{2}m$

$\frac{9}{4}m$

368. A lead shot of 1 mm diameter falls through a long column of glycerine. The variation of its velocity v with distance covered is represented by

369. Two mercury drops (each of radius ‘r’) merge to form bigger drop. The surface energy of the bigger drop, if T is the surface tension, is :

$4\pi r^{2}T$

$2\pi r^{2}T$

$2^{8/3}\pi r^{2}T$

$2^{5/3}\pi r^{2}T$

370. Wax is coated on the inner wall of a capillary tube and the tube is then dipped in water. Then, compared to the unwaxed capillary, the angle of contact $\theta$ and the height h upto which water rises change. These changes are :

$\theta$ increases and h also increases

$\theta$ decreases and h also decreases

$\theta$ increases and h decreases

$\theta$ decreases and h increases

371. A rain drop of radius 0.3 mm has a terminal velocity in air = 1 m/s. The viscosity of air is 8 × 10^–5 poise. The viscous force on it is

45.2 × 10^–4 dyne

101.73×10^–5 dyne

16.95 × 10^–4 dyne

16.95 × 10^–5 dyne

372. A water tank of height 10m, completely filled with water is placed on a level ground. It has two holes one at 3 m and the other at 7 m from its base. The water ejecting from

both the holes will fall at the same spot

upper hole will fall farther than that from the lower hole

upper hole will fall closer than that from the lower hole

more information is required

373. Two capillary of length L and 2L and of radius R and 2R are connected in series. The net rate of flow of fluid through them will be (given rate to the flow through single capillary, $X = \frac{\pi PR^{4}}{8 \eta L}$ )

$\frac{8}{9}X$

$\frac{9}{8}X$

$\frac{5}{7}X$

$\frac{7}{5}X$

374. A candle of diameter d is floating on a liquid in a cylindrical container of diameter D (D >> d) as shown in figure. If it is burning at the rate of 2 cm/hour then the top of the candle will