NEET Physics Solutions For Class 11 Circular Motion Multiple Choice Question

Circular Motion Multiple Choice Question And Answers

Question 1. Two racing cars of masses m₁ and m₂ are moving in circles of radii r₁ and r₂ respectively; their speeds are such that they each make a complete circle in the same time t. The ratio of the angular speed of the first to the second car is :

1. m₁: m₂
2. r₁: r₂
3. 1: 1
4. m₁r₁: m₂r₂

Answer: 1. m₁: m₂

Question 2. A wheel is at rest. Its angular velocity increases uniformly and becomes 80 radians per second after 5 seconds. The total angular displacement is :

1. 800 rad
2. 400 rad
3. 200 rad
4. 100 rad

Answer: 3. 200 rad

Question 3. When a particle moves in a circle with a uniform speed

1. Its velocity and acceleration are both constant
2. Its velocity is constant but the acceleration changes
3. Its acceleration is constant but the velocity changes
4. Its velocity and acceleration both change

Answer: 4. Its velocity and acceleration both change

Question 4. The relation between an angular velocity, the position vector and the linear velocity of a particle moving in a circular path is.

1. \(\vec{\omega} \times \overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{v}}\)
2. \(\vec{\omega} \cdot \vec{r}=\vec{v}\)
3. \(\overrightarrow{\mathbf{r}} \times \vec{\omega}=\overrightarrow{\mathrm{v}}\)
4. \(\vec{\omega} \overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{v}}\)

Answer: 1. \(\vec{\omega} \times \overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{v}}\)

Question 5. A wheel is of diameter 1m. If it makes 30 revolutions/sec., then the linear speed of a point on its circumference will be.

1. 30 π m/s
2. π m/s
3. 60π m/s
4. π/2 m/s

Answer: 1. 30π m/s

Question 6. In a uniform circular motion

1. Both the angular velocity and the angular momentum vary
2. The angular velocity varies but the angular momentum remains constant.
3. Both the angular velocity and the angular momentum stay constant
4. The angular momentum varies but the angular velocity remains constant.

Answer: 3. Both the angular velocity and the angular momentum stay constant

Question 7. The angular speed of a flywheel making 120 revolutions/minute is.

1. 2π rad/s
2. 4π2 rad/s
3. π rad/s
4. 4π rad/s

Answer: 4. 4π rad/s

Question 8. The angular velocity of the second needle in a watch is-

1. \(\frac{\pi}{30}\)
2. 2π
3. π
4. \(\frac{60}{\pi}\)

Answer: 1. \(\frac{\pi}{30}\)

Question 9. The average acceleration vector for a particle having a uniform circular motion is- 

1. A constant vector of magnitude \(\frac{v^2}{r}\)
2. A vector of magnitude \(\frac{v^2}{r}\) directed normal to the plane of the given uniform circular motion.
3. Equal to the instantaneous acceleration vector at the start of the motion.
4. A null vector.

Answer: 4. A null vector.

Question 10. The angular velocity of the minute hand of a clock is:

1. \(\frac{\pi}{30} \mathrm{rad} / \mathrm{s} \)
2. π rad/s
3. 2π rad/s
4. \(\frac{\pi}{1800} \mathrm{rad} / \mathrm{s}\)

Answer: 4. \(\frac{\pi}{1800} \mathrm{rad} / \mathrm{s}\)

Question 11. The second hand of a watch has a length of 6 cm. The speed of the endpoint and magnitude of the difference of velocities at two perpendicular positions will be :

1. 2π and 0 mm/s
2. \(2 \sqrt{2}\) π and 4.44 mm/s
3. \(2 \sqrt{2}\) π and 2π mm/s
4. 2π and \(2 \sqrt{2}\) π mm/s

Answer: 4. 2π and \(2 \sqrt{2}\) π mm/s

Question 12. An aeroplane revolves in a circle above the surface of the earth at a fixed height with a speed of 100 km/hr. The change in velocity after completing 1/2 revolution will be.

1. 200 km/hr
2. 150 km/hr
3. 300 km/hr
4. 400 km/hr

Answer: 1. 200 km/hr

Question 13. A particle moving on a circular path travels the first one-third part of the circumference in 2 sec and the next one-third part in 1 sec. The average angular velocity of the particle is (in rad/sec) –

1. \(\frac{2 \pi}{3}\)
2. \(\frac{\pi}{3}\)
3. \(\frac{4 \pi}{9}\)
4. \(\frac{5 \pi}{3}\)

Answer: 3. \(\frac{4 \pi}{9}\)

Question 14. A grind-stone starts revolving from rest, if its angular acceleration is 4.0 rad/sec² (uniform) then after 4 sec. What are its angular displacement and angular velocity respectively –

1. 32 rad, 16 rad/sec
2. 16 rad, 32 rad/sec
3. 64 rad, 32 rad/sec
4. 32 rad, 64 rad/sec

Answer: 1. 32 rad, 16 rad/sec

Question 15. Angular displacement of any particle is given θ = ω₀t +\(\frac{1}{2}\)αt² where ω₀ and α are constant if ω₀ = 1 rad/sec, α = 1.5 rad/sec² then in t = 2 sec. angular velocity will be (in rad/sec)

1. 1
2. 5
3. 3
4. 4

Answer: 4. 4

Question 16. A particle of mass M is revolving along a circle of radius R and another particle of mass m is revolving in a circle of radius r. If the periods of both particles are the same, then the ratio of their angular velocities is:

1. 1
2. \(\frac{R}{r}\)
3. \(\frac{r}{R}\)
4. \(\sqrt{\frac{R}{r}}\)

Answer: 1. 1

Question 17. In a uniform circular motion

1. Velocity and acceleration remain constant
2. Kinetic energy remains constant
3. Speed and acceleration changes
4. Only velocity changes and acceleration remains constant

Answer: 2. Kinetic energy remains constant

Question 18. Which of the following statements is false for a particle moving in a circle with a constant angular speed?

1. The velocity vector is tangent to the circle
2. The acceleration vector is tangent to the circle
3. The acceleration vector points to the center of the circle
4. The velocity and acceleration vectors are perpendicular to each other

Answer: 2. The acceleration vector is tangent to the circle

Question 19. A particle is acted upon by a force of constant magnitude which is always perpendicular to the velocity of the particle. The motion of the particle takes place in a plane, it follows that

1. Its velocity is constant
2. Its acceleration is constant
3. Its kinetic energy is constant
4. It moves in a straight line

Answer: 3. Its kinetic energy is constant

Question 20. A wheel is subjected to uniform angular acceleration about its axis. Initially, its angular velocity is zero. In the first 2 seconds, it rotates through an angle θ1. In the next 2 sec, it rotates through an additional angle θ \(\frac{\theta_2}{\theta_1}\) is

1. 1
2. 2
3. 3
4. 5

Answer: 3. 3

Question 21. If the equation for the displacement of a particle moving on a circular path is given by (θ)= 2t³ + 0.5, where θ is in radians and t in seconds, then the angular velocity of the particle after 2 sec from its start is

1. 8 rad/sec
2. 12 rad/sec
3. 24 rad/sec
4. 36 rad/sec

Answer: 3. 24 rad/sec

Question 22. For a particle in a non-uniform accelerated circular motion

1. Velocity is radial and acceleration is transverse only
2. Velocity is transverse and acceleration is radial only
3. Velocity is radial and acceleration has both radial and transverse components
4. Velocity is transverse and acceleration has both radial and transverse components

Answer: 4. Velocity is transverse and acceleration has both radial and transverse components

Question 23. Two particles P and Q are located at distances rP and rQ respectively from the axis of a rotating disc such that rP > rQ :

1. Both P and Q have the same acceleration
2. Both P and Q do not have any acceleration
3. P has greater acceleration than Q
4. Q has greater acceleration than P

Answer: 3. P has greater acceleration than Q

Question 24. Let ar and at represent radial and tangential acceleration. The motion of a particle may be circular if :

1. a_r = 0, a_t = 0
2. a_r = 0, a_t ≠ 0
3. a_r ≠ 0, a_t = 0
4. None of these

Answer: 3. a_r ≠ 0, a_t = 0

Question 25. A stone tied to one end of a string 80 cm long is whirled in a horizontal circle with a constant speed. If a stone makes 25 revolutions in 14 seconds, the magnitude of the acceleration of the stone is :

1. 850 cm/s²
2. 996 cm/s²
3. 720 cm/s²
4. 650 cm/s²

Answer: 2. 996 cm/s²

Question 26. A body is moving in a circular path with acceleration a. If its velocity gets doubled, find the ratio of acceleration after and before the change :

1. 1: 4
2. 4: 1
3. 2: 1
4. 2: 1

Answer: 2. 4: 1

Question 27. A spaceman in training is rotated in a seat at the end of a horizontal arm of length 5m. If he can withstand acceleration upto 9 g then what is the maximum number of revolutions per second permissible? (Take g = 10 m/s²)

1. 13.5 rev/s
2. 1.35 rev/s
3. 0.675 rev/s
4. 6.75 rev/s

Answer: 3. 0.675 rev/s

Question 28. A particle of mass m is moving in a uniform circular motion. The momentum of the particle is

1. Constant over the entire path
2. Constantly changes and direction of change is along the tangent
3. Constantly changes and direction of change are along the radial direction
4. Constantly change and direction of change are along a direction which is the instantaneous vector sum of the radial and tangential direction

Answer: 3. Constantly changes and direction of change is along the radial direction

Question 29. A particle is going in a uniform helical and spiral path separately as shown in the figure with constant speed.

1. The velocity of the particle is constant in both cases
2. The acceleration of the particle is constant in both cases
3. The magnitude of acceleration is constant in (1) and decreasing in (2)
4. The magnitude of acceleration is decreasing continuously in both cases

Answer: 3. The magnitude of acceleration is constant in (1) and decreasing in (2)

Question 30. A car is traveling with linear velocity v on a circular road of radius r. If the speed is increasing at the rate of ‘a’ meter/sec², then the resultant acceleration will be –

1. \(\sqrt{\left[\frac{v^2}{r^2}-a^2\right]}\)
2. \(\sqrt{\left.\frac{v^4}{r^2}+a^2\right]}\)
3. \(\sqrt{\left[\frac{v^4}{r^2}-a^2\right]}\)
4. \(\sqrt{\left.\frac{v^2}{r^2}+a^2\right]}\)

Answer: 2. \(\sqrt{\left.\frac{v^4}{r^2}+a^2\right]}\)

Question 31. If the mass, speed & radius of rotation of a body moving on a circular path are increased by 50% then to keep the body moving in a circular path increase in force required will be –

1. 225%
2. 125%
3. 150%
4. 100%

Answer: 2. 125%

Question 32. A motorcycle driver doubles its velocity when he is having a turn. The force exerted outwardly will be.

1. Double
2. Half
3. 4 times
4. 1/4 times

Answer: 3. 4 times

Question 33. For a particle in circular motion, the centripetal acceleration is

1. Less than its tangential acceleration
2. Equal to its tangential acceleration
3. More than its tangential acceleration
4. May be more or less than its tangential acceleration

Answer: 4. May be more or less than its tangential acceleration

Question 34. If the radii of circular paths of two particles of the same masses are in the ratio of 1: 2, then in order to have the same centripetal force, their speeds should be in the ratio of:

1. 1: 4
2. 4: 1
3. 1 : \(\sqrt{2}\)
4. \(\sqrt{2}\): 1

Answer: 3. 1 : \(\sqrt{2}\)

Question 35. On a horizontal smooth surface, a mass of 2 kg is whirled in a horizontal circle by means of a string at an initial angular speed of 5 revolutions per minute. Keeping the radius constant the tension in the string is doubled. The new angular speed is near:

1. 14 rpm
2. 10 rpm
3. 2.25 rpm
4. 7 rpm

Answer: 4. 7 rpm

Question 36. If ar and at represent radial and tangential accelerations, the motion of a particle will be uniformly circular if

1. a_r = 0 and a_t = 0
2. a_r = 0 but a_t ≠ 0
3. a_r ≠ 0 but a_t = 0
4. a_r ≠ 0 and a_t ≠ 0

Answer: 3. a_r ≠ 0 but a_t = 0

Question 37. A string breaks if its tension exceeds 10 newtons. A stone of mass 250 gm tied to this string of length 10 cm is rotated in a horizontal circle. The maximum angular velocity of rotation can be.

1. 20 rad/s
2. 40 rad/s
3. 100 rad/s
4. 200 rad/s

Answer: 1. 20 rad/s

Question 38. A particle moving along a circular path due to a centripetal force having constant magnitude is an example of motion with :

1. Constant speed and velocity
2. Variable speed and velocity
3. Variable speed and constant
4. Velocity constant speed and variable velocity.

Answer: 4. Velocity constant speed and variable velocity.

Question 39. A stone of mass 0.5 kg tied with a string of length 1 meter is moving in a circular path with a speed of 4 m/sec. The tension acting on the string in Newton is –

1. 2
2. 8
3. 0.2
4. 0.8

Answer: 2. 8

Question 40. The formula for centripetal acceleration in a circular motion is.

1. \(\vec{\alpha} \times \overrightarrow{\mathbf{r}}\)
2. \(\vec{\omega} \times \overrightarrow{\mathrm{V}}\)
3. \(\vec{\alpha} \times \overrightarrow{\mathrm{V}}\)
4. \(\vec{\omega} \times \overrightarrow{\mathbf{r}}\)

Answer: 2. \(\vec{\omega} \times \overrightarrow{\mathrm{V}}\)

Question 41. A stone is moved around a horizontal circle with a 20 cm long string tied to it. If centripetal acceleration is 9.8 m/sec², then its angular velocity will be

1. 7 rad/s
2. 22/7 rad/s
3. 49 rad/s
4. 14 rad/s

Answer: 1. 7 rad/s

Question 42. A particle of mass m is executing a uniform motion along a circular path of radius r. If the magnitude of its linear momentum is p, the radial force acting on the particle will be.

1. pmr
2. rm/p
3. mp²/r
4. p²/mr

Answer: 4. p²/mr

Question 43. A particle moves in a circular orbit under the action of a central attractive force inversely proportional to the distance ‘r’. The speed of the particle is.

1. Proportional to r²
2. Independent of r
3. Proportional to r
4. Proportional to 1/r

Answer: 2. Independent of r

Question 44. A particle of mass m is moving in a horizontal circle of radius r under a centripetal force equal to –k/r². The total kinetic energy of the particle is-

1. –k/r
2. k/r
3. k/2r
4. –k/2r

Answer: 3. k/2r

Question 45. A 500 kg car takes around a turn of radius of 50 m with a speed of 36 km/hr. The centripetal force acting on the car will be :

1. 1200 N
2. 1000 N
3. 750 N
4. 250 N

Answer: 2. 1000 N

Question 46. If the radii of circular paths of two particles of the same masses are in the ratio of 1: 2, then in order to have the same centripetal force, their speeds should be in the ratio of :

1. 1: 4
2. 4: 1
3. 1 : \(\sqrt{2}\)
4. \(\sqrt{2}\) : 1

Answer: 3. 1 : \(\sqrt{2}\)

Question 47. A particle is moving in a horizontal circle with constant speed. It has constant

1. Velocity
2. Acceleration
3. Kinetic energy
4. Displacement

Answer: 3. Kinetic energy

Question 48. A particle P will be equilibrium inside a hemispherical bowl of radius 0.5 m at a height 0.2 m from the bottom when the bowl is rotated at an angular speed (g = 10 m/sec²)-

1. \(10 / \sqrt{3} \mathrm{rad} / \mathrm{sec}\)
2. \(10 \sqrt{3} \mathrm{rad} / \mathrm{sec}\)
3. 10 rad/sec
4. \(\sqrt{20} \mathrm{rad} / \mathrm{sec}\)

Answer: 1. \(10 / \sqrt{3} \mathrm{rad} / \mathrm{sec}\)

Question 49. Three identical particles are joined together by a thread. All the three particles are moving on a smooth horizontal plane about point O. If the speed of the outermost particle is v₀, then the ratio of tensions in the three sections of the string is : (Assume that the string remains straight)

1. 3: 5: 7
2. 3: 4: 5
3. 7 : 11: 6
4. 3: 5: 6

Answer: 4. 3: 5: 6

Question 50. A heavy and big sphere is hanging with a string of length l, this sphere moves in a horizontal circular path making an angle θ with vertical then its time period is –

1. \(T=2 \pi \sqrt{\frac{\ell}{g}}\)
2. \(\mathrm{T}=2 \pi \sqrt{\frac{\ell \sin \theta}{\mathrm{g}}}\)
3. \(\mathrm{T}=2 \pi \sqrt{\frac{\cos \theta}{g}}\)
4. \(\mathrm{T}=2 \pi \sqrt{\frac{\ell}{g \cos \theta}}\)

Answer: 3. \(\mathrm{T}=2 \pi \sqrt{\frac{\cos \theta}{g}}\)

Question 51. A gramophone recorder rotates at an angular velocity of ω a coin is kept at a distance r from its center. If μ is static friction constant then the coil will rotate with gramophone if –

1. r > μ g > ω²
2. r = μ g/ω² only
3. r < μ g/ω²
4. r ≤ μ g/ω²

Answer: 4. r ≤ μ g/ω²

Question 52. A train A runs from east to west and another train B of the same mass runs from west to east at the same speed along the equator. A presses the track with a force F₁ and B presses the track with a force F₂.

1. F₁ > F₂
2. F₁ < F₂
3. F₁= F₂
4. The information is insufficient to find the relation between F₁ and F₂.

Answer: 1. F₁ > F₂

Question 53. A cyclist is moving on a circular track of radius 80 m with a velocity of 72 km/hr. He has to lean from the vertical approximately through an angle –

1. tan^-1(1/4)
2. tan^-1(1)
3. tan^-1(1/2)
4. tan^-1(2)

Answer: 3. tan^-1(1/2)

Question 54. A car of mass m is taking a circular turn of radius ‘r’ on a fictional level road with a speed v. In order that the car does not skid –

1. \(\frac{\mathrm{mv}^2}{\mathrm{r}} \geq \mu \mathrm{mg}\)
2. \(\frac{m v^2}{r} \leq \mu \mathrm{mg}\)
3. \(\frac{m v^2}{r}=\mu \mathrm{mg}\)
4. \(\frac{v}{r}=\mu \mathrm{mg}\)

Answer: 2. \(\frac{m v^2}{r} \leq \mu \mathrm{mg}\)

Question 55. A car travels at constant speed on a circular road on level ground. In the figure shown, Fair is the force of air resistance on the car. Which of the other forces best represents the horizontal force of the road on the car’s tires?

1. F_A
2. F_B
3. F_C
4. F_D

Answer: 2. F_B

Question 56. The driver of a car traveling at full speed suddenly sees a wall a distance r directly in front of him. To avoid a collision,

1. He should apply brakes sharply
2. He should turn the car sharply
3. He should apply brakes and then sharply turn
4. None of these

Answer: 1. He should apply brakes sharply

Question 57. A mass is supported on a frictionless horizontal surface. It is attached to a string and rotates about a fixed center at an angular velocity ω₀. If the length of the string and angular velocity are doubled, the tension in the string which was initially T₀0 is now –

1. T₀
2. T₀/2
3. 4T₀
4. 8T₀

Answer: 4. 8T₀

Question 58. Two masses M and m are attached to a vertical axis by weightless threads of combined length l. They are set in rotational motion in a horizontal plane about this axis with constant angular velocity ω. If the tensions in the threads are the same during motion, the distance of M from the axis is.

1. \(\frac{\mathrm{M} \ell}{\mathrm{M}+\mathrm{m}}\)
2. \(\frac{\mathrm{m} \ell}{\mathrm{M}+\mathrm{m}}\)
3. \(\frac{M+m}{M} \ell\)
4. \(\frac{\mathrm{M}+\mathrm{m}}{\mathrm{m}} \ell\)

Answer: 2. \(\frac{\mathrm{m} \ell}{\mathrm{M}+\mathrm{m}}\)

Question 59. A stone tied to the end of a string 1 m long is whirled in a horizontal circle with a constant speed. If the stone makes 22 revolutions in 44s, what is the magnitude and direction of acceleration of the stone?

1. \(\frac{\pi^2}{4} \mathrm{~ms}^{-2}\) and direction along the radius towards the centre 4
2. π²ms^-2 and direction along the radius away from the center
3. π²ms^-2 and direction along the radius towards the center
4. π²ms^-2 and direction along the tangent to the circle

Answer: 3. π²ms^-2 and direction along the radius towards the center

Question 60. The maximum velocity (in ms^-1) with which a car driver can traverse a flat curve of radius 150 m and coefficient of friction 0.6 to avoid skidding is :

1. 60
2. 30
3. 15
4. 25

Answer: 2. 30

Question 61. A cylindrical vessel partially filled with water is rotated about its vertical central axis. It’s surface will

1. Rise equally
2. Rise from the sides
3. Rise from the middle
4. Lowered equally

Answer: 2. Rise from the sides

Question 62. A long horizontal rod has a bead that can slide along its length and is initially placed at a distance L from one end A of the rod. The rod is set in angular motion about A with a constant angular acceleration, α. If the coefficient of friction between the rod and the bead is μ, and gravity is neglected, then the time after which the bead starts slipping is-

1. \(\sqrt{\frac{\mu}{\alpha}}\)
2. \(\frac{\mu}{\sqrt{\alpha}}\)
3. \(\frac{1}{\sqrt{\mu \alpha}}\)
4. Infinitesimal

Answer: 1. \(\sqrt{\frac{\mu}{\alpha}}\)

Question 63. A ball of mass (m) 0.5 kg is attached to the end of a string having length (L) 0.5 m. The ball is rotated on a horizontal circular path about a vertical axis. The maximum tension that the string can bear is 324 N. The maximum possible value of the angular velocity of the ball (in radian/s) is:

1. 9
2. 18
3. 27
4. 36

Answer: 4. 36

Question 64. A particle of mass m is moving with constant velocity \(\overrightarrow{\mathrm{V}}\) on a smooth horizontal surface. A constant force starts acting on a particle perpendicular to velocity v. The Radius of curvature after force F starts acting is:

1. \(\frac{m v^2}{F}\)
2. \(\frac{m v^2}{F \cos \theta}\)
3. \(\frac{m v^2}{F \sin \theta}\)
4. None of these

Answer: 1. \(\frac{m v^2}{F}\)

Question 65. A stone is projected with speed u and the angle of projection is θ. Find the radius of curvature at t = 0.

1. \(\frac{u^2 \cos ^2 \theta}{g}\)
2. \(\frac{u^2}{g \sin \theta}\)
3. \(\frac{u^2}{g \cos \theta}\)
4. \(\frac{u^2 \sin ^2 \theta}{g}\)

Answer: 3. \(\frac{u^2}{g \cos \theta}\)

Question 66. The velocity and acceleration vectors of a particle undergoing circular motion are \(\overrightarrow{\mathrm{v}}=2 \hat{\mathrm{i}} \mathrm{m} / \mathrm{s}\) and \(\vec{a}=2 \hat{i}+4 \hat{j} \mathrm{~m} / \mathrm{s}^2\) respectively at an instant of time. The radius of the circle is

1. 1m
2. 2m
3. 3m
4. 4m

Answer: 1. 1m

Question 67. A particle is projected horizontally from the top of a tower with a velocity v0. If v is its velocity at any instant, then the radius of curvature of the path of the particle at that instant is directly proportional to:

1. v³
2. v²
3. v
4. 1/v

Answer: 1. 1/v

Question 68. The tension in the string revolving in a vertical circle with a mass m at the end when it is at the lowest position.

1. \(\frac{m v^2}{r}\)
2. \(\frac{m v^2}{r}-m g\)
3. \(\frac{m v^2}{r}+m g\)
4. mg

Answer: 3. \(\frac{m v^2}{r}+m g\)

Question 69. A motorcycle is going on an overbridge of radius R. The driver maintains a constant speed. As the motorcycle is ascending on the overbridge, the normal force on it :

1. Increase
2. Decreases
3. Remains constant
4. First increases then decreases.

Answer: 1. Increase

Question 70. In a circus, a stuntman rides a motorbike in a circular track of radius R in the vertical plane. The minimum speed at the highest point of the track will be :

1. \(\sqrt{2 \mathrm{gR}}\)
2. 2gR
3. \(\sqrt{3 \mathrm{gR}}\)
4. \(\sqrt{g R}\)

Answer: 4. \(\sqrt{g R}\)

Question 71. A particle is moving in a vertical circle. The tensions in the string when passing through two positions at angles 30° and 60° from vertical (lowest positions) are T₁ and T₂ respectively. Then

1. T₁ = T₂
2. T₂ > T₁
3. T₁ > T₂
4. Tension in the string always remains the same

Answer: 3. T₁ > T₂

Question 72. A car moves at a constant speed on a road. The normal force by the road on the car is N_A and N_B when it is at points A and B respectively.

1. N_A = N_B
2. N_A > N_B
3. N_A < N_B
4. Insufficient

Answer: 2. N_A > N_B

Question 73. A heavy mass is attached to a thin wire and is whirled in a vertical circle. The wire is most likely to break.

1. When the mass is at the height point of the circle
2. When the mass is at the lowest point of the circle
3. When the wire is horizontal
4. At an angle of cos^-1(1/3) from the upward vertical

Answer: 2. When the mass is at the lowest point of the circle

Question 74. A hollow sphere has a radius of 6.4 m. The minimum velocity required by a motorcyclist at the bottom to complete the circle will be.

1. 17.7 m/s
2. 10.2 m/s
3. 12.4 m/s
4. 16.0 m/s

Answer: 1. 17.7 m/s

Question 75. A body of mass 100 g is rotating in a circular path of radius r with constant speed. The work done in one complete revolution is.

1. 100 rJ
2. (r/100) J
3. (100/r) J
4. Zero “

Answer: 4. Zero “

Question 76. A weightless thread can bear tension upto 3.7 kg wt. A stone of mass 500 gms is tied to it and revolved in a circular path of radius 4 m in a vertical plane. If g = 10 ms^-2, then the maximum angular velocity of the stone will be.

1. 4 radians/sec
2. 16 radians/sec
3. 21radians/sec
4. 2 radians/sec

Answer: 1. 4 radians/sec

Question 77. A small disc is on the top of a hemisphere of radius R. What is the smallest horizontal velocity v that should be given to the disc for it to leave the hemisphere and not slide down it? [There is no friction]

1. \(v=\sqrt{2 g R}\)
2. \(v=\sqrt{g R}\)
3. \(v=\frac{g}{R}\)
4. \(v=\sqrt{g^2 R}\)

Answer: 2. \(v=\sqrt{g R}\)

Question 78. The maximum velocity at the lowest point, so that the string just slacks at the highest point in a vertical circle of radius l.

1. \(\sqrt{g \ell}\)
2. \(\sqrt{3 \mathrm{~g} \ell}\)
3. \(\sqrt{5 \mathrm{~g} \ell}\)
4. \(\sqrt{7 g \ell}\)

Answer: 3. \(\sqrt{5 \mathrm{~g} \ell}\)

Question 79. A simple pendulum oscillates in a vertical plane. When it passes through the mean position, the tension in the string is 3 times the weight of the pendulum bob. What is the maximum displacement of the pendulum of the string with respect to the vertical?

1. 30º
2. 45º
3. 60º
4. 90º

Answer: 4. 90º

Question 80. A coin placed on a rotating turntable just slips if it is placed at a distance of 4 cm from the center. If the angular velocity of the turntable is doubled, it will just slip at a distance of

1. 1 cm
2. 2 cm
3. 4 cm
4. 8 cm

Answer: 1. 1 cm

Question 81. A cane filled with water is revolved in a vertical circle of radius 4 meter and the water just does not fall down. The time period of the revolution will be

1. 1 sec
2. 10 sec
3. 8 sec
4. 4 sec

Answer: 4. 4 sec

Question 82. A weightless rod of length 2l carries two equal masses ‘m’, one tied at the lower end of A and the other at the middle of the rod at B. The rod can rotate in a vertical plane about a fixed horizontal axis passing through C. The rod is released from rest in a horizontal position. The speed of mass B at the instant rod, becomes vertical is :

1. \(\sqrt{\frac{3 \mathrm{~g} \ell}{5}}\)
2. \(\sqrt{\frac{4 g \ell}{5}}\)
3. \(\sqrt{\frac{6 \mathrm{~g} \ell}{5}}\)
4. \(\sqrt{\frac{7 g \ell}{5}}\)

Answer: 3. \(\sqrt{\frac{6 \mathrm{~g} \ell}{5}}\)

Question 83. A body is suspended from a smooth horizontal nail by a string of length 0.25 meters. What minimum horizontal velocity should be given to it in the lowest position so that it may move in a complete vertical circle with the nail at the center?

1. 3.5 ms^-1
2. 4.9 ms^-1
3. 7\(\sqrt{2}\) ms^-1
4. \(\sqrt{9.8}\) ms^-1

Answer: 1. 3.5 ms^-1

Question 84. A block of mass m slides down along the surface of the bowl from the rim to the bottom as shown in Fig. The velocity of the block at the bottom will be –

1. \(\sqrt{\pi \mathrm{Rg}}\)
2. \(2 \sqrt{\pi \mathrm{Rg}}\)
3. \(\sqrt{2 \mathrm{Rg}}\)
4. \(\sqrt{g R}\)

Answer: 3. \(\sqrt{2 \mathrm{Rg}}\)

Question 85. A mass m is revolving in a vertical circle at the end of a string of length 20 cm. By how many times does the tension of the string at the lowest point exceed the tension at the topmost point –

1. 2 mg
2. 4 mg
3. 6 mg
4. 8 mg

Answer: 3. 6 mg

Question 86. A block follows the path as shown in the figure from height h. If the radius of the circular path is r, then the relation holds well to complete full circle.

1. h < 5r/2
2. h > 5r/2
3. h = 5r/2
4. h ≥ 5r/2

Answer: 4. h ≥ 5r/2

Question 87. A particle is kept at rest at the top of a sphere of diameter 42 m. When disturbed slightly, it slides down. At what height ‘h’ from the bottom, the particle will leave the sphere?

1. 14 m
2. 28 m
3. 35 m
4. 7 m

Answer: 3. 35 m

Question 88. A stone of 1 kg tied up with a 10/3 meter long string rotated in a vertical circle. If the ratio of maximum and minimum tension in the string is 4 then the speed of the stone at the highest point of the circular path will be – (g = 10 m/s²)

1. 20 m/s
2. \(10 \sqrt{3} \mathrm{~m} / \mathrm{s}\)
3. \(5 \sqrt{2} \mathrm{~m} / \mathrm{s}\)
4. 10 m/s

Answer: 4. 10 m/s

Question 89. A child is swinging a swing, Minimum and maximum heights of the swing from the earth’s surface are 0.75 m and 2 m respectively. The maximum velocity of this swing is :

1. 5 m/s
2. 10 m/s
3. 15 m/s
4. 20 m/s

Answer: 1. 5 m/s

Question 90. A stone is tied to a string of length l and is whirled in a vertical circle with the other end of the string as the center. At a certain instant of time, the stone is at its lowest position and has a speed u. The magnitude of the change in velocity as it reaches a position where the string is horizontal (g being acceleration due to gravity) is :

1. \(\sqrt{2\left(\mathrm{u}^2-\mathrm{g} \ell\right)}\)
2. \(\sqrt{\left(u^2-g \ell\right)}\)
3. \(u-\sqrt{\left(u^2-2 g \ell\right)}\)
4. \(\sqrt{2 g \ell}\)

Answer: 1. \(\sqrt{2\left(\mathrm{u}^2-\mathrm{g} \ell\right)}\)

Question 91. In a circus, a stuntman rides a motorbike in a circular track of radius R in the vertical plane. The minimum speed at the highest point of the track will be :

1. \(\sqrt{2 g R}\)
2. 2gR
3. \(\sqrt{3 \mathrm{gR}}\)
4. \(\sqrt{g R}\)

Answer: 4. \(\sqrt{g R}\)

Question 92. A particle of mass m begins to slide down a fixed smooth sphere from the top. What is its tangential acceleration when it breaks off the sphere?

1. \(\frac{2 \mathrm{~g}}{3}\)
2. \(\frac{\sqrt{5} g}{3}\)
3. g
4. \(\frac{g}{3}\)

Answer: 2. \(\frac{\sqrt{5} g}{3}\)

Question 93. A body of mass 1 kg is moving in a vertical circular path of radius 1m. The difference between the kinetic energies at its highest and lowest position is

1. 20J
2. 10J
3. \(4 \sqrt{5} \mathrm{~J}\)
4. \(10(\sqrt{5}-1) \mathrm{J}\)

Answer: 1. 20J

Question 94. A small block is shot into each of the four tracks as shown below. Each of the tracks rises to the same height. The speed with which the block enters the track is the same in all cases. At the highest point of the track, the normal reaction is maximum in –

Answer: 1.

Question 95. A simple pendulum is oscillating without damping. When the displacements of the bob is less than maximum, its acceleration vector \(\overrightarrow{\mathrm{a}}\) is correctly shown in

Answer: 3.

Question 96. A car moving on a horizontal road may be thrown out of the road in taking a turn:

1. By the gravitational force
2. Due to a lack of sufficient centripetal force
3. Due to friction between the road and the tire
4. Due to the reaction of the earth

Answer: 2. Due to a lack of sufficient centripetal force

Question 97. The magnitude of the centripetal force acting on a body of mass m executing uniform motion in a circle of radius r with speed υ is-

1. mυr
2. \(\frac{m v^2}{r}\)
3. \(\frac{v}{r^2 m}\)
4. \(\frac{v}{\mathrm{rm}}\)

Answer: 2. \(\frac{m v^2}{r}\)

Question 98. The radius of the curved road on the national highway is R. The Width of the road is b. The outer edge of the road is raised by h with respect to the inner edge so that a car with velocity υ can pass safely over it. The value of h is-

1. \(\frac{v^2 b}{R g}\)
2. \(\frac{v}{R g b}\)
3. \(\frac{v^2 R}{g}\)
4. \(\frac{u^2 b}{R}\)

Answer: 1. \(\frac{v^2 b}{R g}\)

Question 99. If the apparent weight of the bodies at the equator is to be zero, then the earth should rotate with angular velocity

1. \(\sqrt{\frac{\mathrm{g}}{\mathrm{R}}} \mathrm{rad} / \mathrm{sec}\)
2. \(\sqrt{\frac{2 g}{R}} \mathrm{rad} / \mathrm{sec}\)
3. \(\sqrt{\frac{\mathrm{g}}{2 \mathrm{R}}} \mathrm{rad} / \mathrm{sec}\)
4. \(\sqrt{\frac{3 \mathrm{~g}}{2 \mathrm{R}}} \mathrm{rad} / \mathrm{sec}\)

Answer: 1. \(\sqrt{\frac{\mathrm{g}}{\mathrm{R}}} \mathrm{rad} / \mathrm{sec}\)

Question 100. The road is 10 m wide. Its radius of curvature is 50 m. The outer edge is above the lower edge by a distance of 1.5 m. This road is most suited for the velocity

1. 2.5 m/sec
2. 4.5 m/sec
3. 6.5 m/sec
4. 8.5 m/sec

Answer: 4. 6.5 m/sec

Question 101. The radius of the curved road on the national highway is R. The Width of the road is b. The outer edge of the road is raised by h with respect to the inner edge so that a car with velocity v can pass safely over it. The value of h is

1. \(\frac{v^2 b}{R g}\)
2. \(\frac{\mathrm{v}}{\mathrm{Rgb}}\)
3. \(\frac{v^2 R}{g}\)
4. \(\frac{v^2 b}{R}\)

Answer: 1. \(\frac{v^2 b}{R g}\)

Question 102. A circular road of radius 1000 m has a banking angle of 45º. The maximum safe speed of a car having a mass of 2000 kg will be if the coefficient of friction between tire and road is 0.5

1. 172 m/s
2. 124 m/s
3. 99 m/s
4. 86 m/s

Answer: 1. 172 m/s

Question 103. A cane filled with water is revolved in a vertical circle of radius 4 meter and the water just does not fall down. The time period of the revolution will be

1. 1 sec
2. 10 sec
3. 8 sec
4. 4 sec

Answer: 4. 4 sec

Question 104. A motorcyclist moving with a velocity of 72 km/hour on a flat road takes a turn on the road at a point where the radius of curvature of the road is 20 meters. The acceleration due to gravity is 10 m/sec². In order to avoid skidding, he must not bend with respect to the vertical plane by an angle greater than-

1. θ = tan^-1 6
2. θ = tan^-1 2
3. θ = tan^-1 25.92
4. θ = tan^-1 4

Answer: 2. θ = tan^-1 2

Question 105. The kinetic energy k of a particle moving along a circle of radius R depends on the distance covered s as k = as2 where a is a constant. The force acting on the particle is

1. \(2 a \frac{s^2}{R}\)
2. \({2as}\left(1+\frac{s^2}{R^2}\right)^{1 / 2}\)
3. 2as
4. \(2 \mathrm{a} \frac{\mathrm{R}^2}{\mathrm{~s}}\)

Answer: 2. \({2as}\left(1+\frac{s^2}{R^2}\right)^{1 / 2}\)

Question 106. A particle of mass m is moving in a circular path of constant radius r such that its centripetal acceleration ac is varying with time t as ac = k²rt² where k is a constant. The power delivered to the particle by the force acting on it is-

1. 2πmk²r²
2. mk²r²t
3. \(\frac{\left(m k^4 r^2 t^5\right)}{3}\)
4. Zero

Answer: 2. mk²r²t

Question 107. A small block slides with velocity 0.5 gron the horizontal frictionless surface as shown in the Figure. The block leaves the surface at point C. The angle θ in the Figure is :

1. cos^-1(4/9)
2. cos^-1(3/4)
3. cos^-1(1/2)
4. None of the above

Answer: 4. None of the above

Question 108. A particle moves along a circle of radius \(\left(\frac{20}{\pi}\right)\) with constant tangential acceleration. If the speed of the particle is 80 m/s at the end of the second revolution after motion has begun, the tangential acceleration is:

1. 160 π m/s²
2. 40 π m/s²
3. 40 m/s²
4. 640 π m/s²

Answer: 3. 40 m/s²

Question 109. Centrifugal force is an inertial force when considered by –

1. An observer at the center of circular motion
2. An outside observer
3. An observer who is moving with the particle that is experiencing the force
4. None of the above

Answer: 3. An observer who is moving with the particle which is experiencing the force

Question 110. A rod of length L is pivoted at one end and is rotated with a uniform angular velocity in a horizontal plane. Let T₁ and T₂ be the tensions at the points L/4 and 3L/4 away from the pivoted ends.

1. T₁ > T₂
2. T₂ > T₁
3. T₁ = T₂
4. The relation between T₁ and T₂ depends on whether the rod rotates clockwise or anticlockwise

Answer: 1. T₁ > T₂

Question 111. When a ceiling fan is switched off its angular velocity reduces to 50% while it makes 36 rotations. How many more rotations will it make before coming to rest (Assume uniform angular retardation)

1. 18
2. 12
3. 36
4. 48

Answer: 2. 12

Question 112. A particle is moving in the vertical plane. It is attached at one end of a string of length l whose other end is fixed. The velocity at the lowest point is u. The tension in the string \(\overrightarrow{\mathrm{T}}\) is and acceleration of the particle \(\overrightarrow{\mathrm{a}}\) is at any position. Then, \(\overrightarrow{\mathrm{T}}.\overrightarrow{\mathrm{a}}\) is zero at the highest point:

1. Only if \(u \leq \sqrt{2 \mathrm{~g} \ell}\)
2. If \(\sqrt{5 \mathrm{~g} \ell}\)
3. Only if \(\mathrm{u}=\sqrt{2 \mathrm{~g} \ell}\)
4. Only if \(u>\sqrt{2 g \ell}\)

Answer: 2. If \(\sqrt{5 \mathrm{~g} \ell}\)

Question 113. In the above question, \(\overrightarrow{\mathrm{T}} \cdot \overrightarrow{\mathrm{a}}\) T.a is non-negative at the lowest point for:

1. \(\mathrm{u} \leq \sqrt{2 \mathrm{~g} \ell}\)
2. \(\mathrm{u}=\sqrt{2 \mathrm{~g} \ell}\)
3. \(\mathrm{u}<\sqrt{2 \mathrm{~g} \ell}\)
4. Any value of u

Answer: 4. Any value of u

Question 114. In the above question, \(\overrightarrow{\mathrm{T}} . \vec{u}\) is zero for:

1. \(\mathrm{u} \leq \sqrt{2 \mathrm{~g} \ell}\)
2. \(\mathrm{u}=\sqrt{2 \mathrm{~g} \ell}\)
3. \(\mathrm{u} \geq \sqrt{2 \mathrm{~g} \ell}\)
4. Any value of u

Answer: 4. Any value of u

Question 115. A bob of mass M is suspended by a massless string of length L. The horizontal velocity V at position A is just sufficient to make it reach the point B. The angle θ at which the speed of the bob is half of that at A satisfies

1. \(\theta=\frac{\pi}{4}\)
2. \(\frac{\pi}{4}<\theta<\frac{\pi}{2}\)
3. \(\frac{\pi}{2}<\theta<\frac{3 \pi}{4}\)
4. \(\frac{3 \pi}{4}<\theta<\pi\)

Answer: 4. \(\frac{3 \pi}{4}<\theta<\pi\)

Question 116. If a particle of mass m is moving in a horizontal circle of radius r with a centripetal force \(\left(-\frac{\mathrm{K}}{\mathrm{r}^2}\right)\), the total energy is-

1. \(-\frac{\mathrm{K}}{2 \mathrm{r}}\)
2. \(-\frac{\mathrm{K}}{\mathrm{r}}\)
3. \(-\frac{2 \mathrm{~K}}{\mathrm{r}}\)
4. \(-\frac{4 K}{r}\)

Answer: 1. \(-\frac{\mathrm{K}}{2 \mathrm{r}}\)

Question 117. A particle moves in a circle of radius 5 cm with constant speed and time period 0.2 πs. The acceleration of the particle is :

1. 15 m/s²
2. 25 m/s²
3. 36 m/s²
4. 5 m/s²

Answer: 4. 5 m/s²

Question 118. A car of mass 1000 kg negotiates a banked curve of radius 90 m on a frictionless road. If the banking angle is 45º, the speed of the car is :

1. 20 ms^-1
2. 30 ms^-1
3. 5 ms^-1
4. 10 ms^-1

Answer: 2. 30 ms^-1

Question 119. A car of mass m is moving on a level circular track of radius R. If μs represents the static friction between the road and tires of the car, the maximum speed of the car in circular motion is given by :

1. \(\sqrt{\mu_{\mathrm{s}} \mathrm{mRg}}\)
2. \(\sqrt{\mathrm{Rg} / \mu_{\mathrm{s}}}\)
3. \(\sqrt{\mathrm{mRg} / \mu_{\mathrm{s}}}\)
4. \(\sqrt{\mu_{\mathrm{s}} \mathrm{Rg}}\)

Answer: 4. \(\sqrt{\mu_{\mathrm{s}} \mathrm{Rg}}\)

Question 120. Two stones of masses m and 2 m are whirled in horizontal circles the heavier one in radius \(\{r}{2}\) and the lighter one in radius r. The tangential speed of lighter stones is n times that of the value of heavier stones when they experience the same centripetal forces. The value of n is :

1. 3
2. 4
3. 1
4. 2

Answer: 4. 2

Question 121. The position vector of a particle \(\overrightarrow{\mathrm{R}}\) as a function of time is given by:

⇒ \(\vec{R}=4 \sin (2 \pi t) \hat{i}+4 \cos (2 \pi t)\)

Where R is in meters, t is seconds, and \(\hat{i} \text { and } \hat{j}\) denote unit vectors along x-and y-directions, respectively. Which one of the following statements is wrong for the motion of a particle?

1. Magnitude of acceleration vector is \(\frac{v^2}{R}\), where v is the velocity of particle
2. The magnitude of the velocity of the particle is 8 meters/second
3. path of the particle is a circle of radius 4 meters.
4. Acceleration vector is along – \(\vec{R}\)

Answer: 2. Magnitude of the velocity of the particle is 8 meters/second

Question 122. What is the minimum velocity with which a body of mass m must enter a vertical loop of radius R so that it can complete the loop?

1. \(\sqrt{5 \mathrm{gR}}\)
2. \(\sqrt{g R}\)
3. \(\sqrt{2 g R}\)
4. \(\sqrt{3 \mathrm{gR}}\)

Answer: 1. \(\sqrt{5 \mathrm{gR}}\)

Question 123. 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?

1. 0.2 m/s²
2. 0.1 m/s²
3. 0.15 m/s²
4. 0.18 m/s²

Answer: 2. 0.1 m/s²

Question 124. A car is negotiating a curved road of radius R. The road is banked at an angle θ. The coefficient of friction between the tires of the car and the road is μs. The maximum safe velocity on this road is:

1. \(\sqrt{\frac{\mathrm{g}}{\mathrm{R}^2} \frac{\mu_{\mathrm{s}}+\tan \theta}{1-\mu_{\mathrm{s}}+\tan \theta}}\)
2. \(\sqrt{g R^2 \frac{\mu_s+\tan \theta}{1-\mu_s+\tan \theta}}\)
3. \(\sqrt{g R \frac{\mu_s+\tan \theta}{1-\mu_s+\tan \theta}}\)
4. \(\sqrt{\frac{g}{R} \frac{\mu_{\mathrm{s}}+\tan \theta}{1-\mu_{\mathrm{s}}+\tan \theta}}\)

Answer: 3. \(\sqrt{g R \frac{\mu_s+\tan \theta}{1-\mu_s+\tan \theta}}\)

Question 125. In the given figure, a = 15 m/s² represents the total acceleration of a particle moving in the clockwise direction in a circle of radius R = 2.5 m at a given instant of time. The speed of the particle is

1. 6.2 m/s
2. 4.5 m/s
3. 5.0 m/s
4. 5.7 m/s

Answer: 4. 5.7 m/s

Question 126. One end of a string of length l is connected to a particle of mass ‘m’ and the other end is connected to a small peg on a smooth horizontal table. If the particle moves in a circle with speed ‘v’ the net force on the particle (directed towards the center) will be (T represents the tension in the string)

1. T
2. \(\mathrm{T}+\frac{\mathrm{m} \mathrm{v}^2}{\ell}\)
3. \(\mathrm{T}-\frac{\mathrm{m} \mathrm{v}^2}{\ell}\)
4. zero

Answer: 1. T

Question 127. A body initially at rest and sliding along a frictionless track from a height h (as shown in the figure) Just completes a vertical circle of diameter AB = D. The height h is equal to

1. \(\frac{3}{2} D\)
2. \(\frac{5}{4} D\)
3. \(\frac{7}{5} D\)
4. D

Answer: 2. \(\frac{5}{4} D\)

Question 128. A mass m is attached to a thin wire and whirled in a vertical circle. The wire is most likely to break when:

1. Inclined at an angle of 60º from vertical
2. The mass is at the highest point
3. The wire is horizontal
4. The mass is at the lowest point

Answer: 4. The mass is at the lowest point

Question 129. A block of mass 10 kg in contact against the inner wall of a hollow cylindrical drum of radius 1m. The coefficient of friction between the block and the inner wall of the cylinder is 0.1. The minimum angular velocity needed for the cylinder to keep the block stationary when the cylinder is vertical and rotating about its axis will be (g = 10 m/s²)

1. 10 π rad/s
2. \(\sqrt{10} \mathrm{rad} / \mathrm{s}\)
3. \(\frac{10}{2 \pi} \mathrm{rad} / \mathrm{s}\)
4. 10 π rad/s

Answer: 4. 10 π rad/s

Question 130. A particle starting from rest, moves in a circle of radius ‘r’. It attains a velocity of V0 m/s in the nth round. Its angular acceleration will be:

1. \(\frac{V_0}{n} \mathrm{rad} / \mathrm{s}^2\)
2. \(\frac{V_0^2}{2 \pi \mathrm{nr}^2} \mathrm{rad} / \mathrm{s}^2\)
3. \(\frac{V_0^2}{4 \pi \mathrm{r}^2} \mathrm{rad} / \mathrm{s}^2\)
4. \(\frac{\mathrm{V}_0^2}{4 \pi \mathrm{nr}} \mathrm{rad} / \mathrm{s}^2\)

Answer: 3. \(\frac{V_0^2}{4 \pi \mathrm{r}^2} \mathrm{rad} / \mathrm{s}^2\)

Question 131. A po1nt P moves in a 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 meters and t is in seconds. The radius of the path is 20 m. The acceleration of ‘P’ when t = 2 s is nearly.

1. 13 m/s²
2. 12 m/s²
3. 7.2 m/s²
4. 14 m/s²

Answer: 4. 14 m/s²

Question 132. For a particle in uniform circular motion, the acceleration \(\overrightarrow{\mathrm{a}}\) at a point P (R, θ) on the circle of radius R is (Here θ is measured from the x-axis)

1. \(-\frac{v^2}{R} \cos \theta \hat{i}+\frac{v^2}{R} \sin \theta \hat{j}\)
2. \(-\frac{v^2}{R} \sin \theta \hat{i}+\frac{v^2}{R} \cos \theta \hat{j}\)
3. \(-\frac{v^2}{R} \cos \theta \hat{i}-\frac{v^2}{R} \sin \theta \hat{j}\)
4. \(\frac{v^2}{R} \hat{i}+\frac{v^2}{R} \hat{j}\)

Answer: 3. \(-\frac{v^2}{R} \sin \theta \hat{i}+\frac{v^2}{R} \cos \theta \hat{j}\)

Question 133. Two cars of masses m₁ and m2 are moving in circles of radii r₁ and r₂, respectively. Their speeds are such that they make complete circles at the same time t. The ratio of their centripetal acceleration is:

1. m₁ r₁ : m₂r₂
2. m₁ : m₂
3. r₁ : r₂
4. 1: 1

Answer: 4. 1:1

Question 134. A particle is moving with a uniform speed in a circular orbit of radius R in a central force inversely proportional to the n^th power of R. If the period of rotation of the particle is T, then:

1. \(\mathrm{T} \alpha \mathrm{R}^{(\mathrm{n}+1) / 2}\)
2. \(\mathrm{T} \alpha \mathrm{R}^{\mathrm{n} / 2}\)
3. \(\mathrm{T} \alpha \mathrm{R}^{3 / 2}\) For any n
4. \(T \alpha R^{\frac{n}{2}+1}\)

Answer: 1. \(\mathrm{T} \alpha \mathrm{R}^{(\mathrm{n}+1) / 2}\)

Question 135. A particle is moving along a circular path with a constant speed of 10 ms^-1. What is the magnitude of the change in the velocity of the particle, when it moves through an angle of 60° around the center of the circle?

1. Zero
2. 10 m/s
3. \(10 \sqrt{2} \mathrm{~m} / \mathrm{s}\)
4. \(10 \sqrt{3} \mathrm{~m} / \mathrm{s}\)

Answer: 2. 10 m/s

Question 136. Two particles A and, B are moving on two concentric circles of radii R₁ and R₂ with equal angular speed ω. At t = 0, their positions and direction of motion are shown in the figure.

The relative velocity \(\vec{v}_A-\vec{v}_B \text { at } t=\frac{\pi}{2 \omega}\) is

1. \(\omega\left(R_2-R_1\right) \hat{i}\)
2. \(\omega\left(R_1-R_2\right) \hat{i}\)
3. \(-\omega\left(R_1+R_2\right) \hat{i}\)
4. \(\left(R_1+R_2\right) \hat{i}\)

Answer: 1. \(\omega\left(R_2-R_1\right) \hat{i}\)