Here’s a set of 50 MCQs for Unit 5: Motion of System of Particles and Rigid Body, along with explanations for each answer:
1. The center of mass of a system of particles is the point where?
A) The total mass of the system is concentrated.
B) The mass of the system is evenly distributed.
C) The system is in equilibrium.
D) The velocity of the system is zero.
Answer: A) The total mass of the system is concentrated.
Explanation: The center of mass is a point where the entire mass of a system can be considered as concentrated for the purposes of motion analysis.
2. The motion of a rigid body is described by?
A) Only translational motion
B) Only rotational motion
C) Both translational and rotational motion
D) Vibrational motion
Answer: C) Both translational and rotational motion.
Explanation: A rigid body can have both translational (linear) motion and rotational (angular) motion. Most of the motion of real objects involves both components.
3. The moment of inertia of a body is the rotational analog of?
A) Mass
B) Force
C) Velocity
D) Work
Answer: A) Mass.
Explanation: Moment of inertia is the rotational equivalent of mass, as it quantifies the resistance of a body to changes in its rotational motion.
4. Which of the following is not a type of motion of a rigid body?
A) Translational motion
B) Rotational motion
C) Vibrational motion
D) None of the above
Answer: D) None of the above.
Explanation: All the listed motions—translational, rotational, and vibrational—are types of motion for rigid bodies. The motion of a rigid body can be classified into these three types.
5. The center of mass of a body moves as if all the external forces were acting on?
A) The whole mass of the body at the center of mass.
B) The center of mass.
C) The surface of the body.
D) The edges of the body.
Answer: A) The whole mass of the body at the center of mass.
Explanation: The center of mass behaves as though all the external forces are applied at that point, simplifying the analysis of the motion of the entire system.
6. The center of mass of a two-particle system is located at a point?
A) Closer to the heavier particle.
B) Equidistant from both particles.
C) Closer to the lighter particle.
D) At the midpoint between the two particles.
Answer: A) Closer to the heavier particle.
Explanation: The center of mass lies closer to the heavier particle since it is weighted by the mass of the objects in the system.
7. The rotational kinetic energy of a rigid body is given by?
A) 12mv2\frac{1}{2}mv^221mv2
B) 12Iω2\frac{1}{2}I\omega^221Iω2
C) Iω2I\omega^2Iω2
D) mghmghmgh
Answer: B) 12Iω2\frac{1}{2}I\omega^221Iω2.
Explanation: The rotational kinetic energy of a rigid body is 12Iω2\frac{1}{2}I\omega^221Iω2, where III is the moment of inertia and ω\omegaω is the angular velocity.
8. Which of the following is true about the motion of the center of mass?
A) The center of mass moves only under the influence of external forces.
B) The center of mass moves under the influence of internal forces.
C) The center of mass does not move if there is no external force.
D) The center of mass moves in a straight line in all cases.
Answer: A) The center of mass moves only under the influence of external forces.
Explanation: The center of mass moves according to Newton’s second law, under the influence of external forces. Internal forces don’t affect the motion of the center of mass.
9. In the case of a rotating rigid body, angular momentum is given by?
A) IωI\omegaIω
B) mvrmvrmvr
C) 12mv2\frac{1}{2}mv^221mv2
D) IαI\alphaIα
Answer: A) IωI\omegaIω.
Explanation: The angular momentum of a rotating rigid body is given by the product of its moment of inertia III and its angular velocity ω\omegaω.
10. Which of the following quantities is conserved for a system of particles if no external torque is acting on it?
A) Kinetic energy
B) Momentum
C) Angular momentum
D) Force
Answer: C) Angular momentum.
Explanation: If no external torque is applied, angular momentum is conserved in a system of particles, similar to how linear momentum is conserved in the absence of external forces.
11. A rotating body has both?
A) Translational kinetic energy and rotational kinetic energy.
B) Only translational kinetic energy.
C) Only rotational kinetic energy.
D) No kinetic energy.
Answer: A) Translational kinetic energy and rotational kinetic energy.
Explanation: A rotating body can have both translational kinetic energy (due to its motion as a whole) and rotational kinetic energy (due to its rotation around an axis).
12. The inertia of a body is defined as?
A) The resistance to change in the body’s motion.
B) The force required to move a body.
C) The body’s tendency to remain at rest.
D) The body’s tendency to rotate.
Answer: A) The resistance to change in the body’s motion.
Explanation: Inertia is the property of a body to resist any change in its motion, whether that be a change in its velocity (linear motion) or a change in its rotational state.
13. Which of the following describes the motion of a system of particles under the influence of internal forces only?
A) The system will move in a straight line.
B) The system will rotate.
C) The system will have constant velocity.
D) The center of mass will not move.
Answer: D) The center of mass will not move.
Explanation: Internal forces within the system cancel out in terms of their effect on the motion of the center of mass, so the center of mass remains at rest or moves with constant velocity.
14. The torque applied to an object is the rotational analog of?
A) Force
B) Mass
C) Velocity
D) Acceleration
Answer: A) Force.
Explanation: Torque is the rotational equivalent of force. It is responsible for producing rotational motion, just as force causes translational motion.
15. The moment of inertia of a solid disk about an axis through its center and perpendicular to its plane is?
A) 12mr2\frac{1}{2}mr^221mr2
B) 14mr2\frac{1}{4}mr^241mr2
C) mr2mr^2mr2
D) 13mr2\frac{1}{3}mr^231mr2
Answer: A) 12mr2\frac{1}{2}mr^221mr2.
Explanation: The moment of inertia for a solid disk rotating about an axis through its center and perpendicular to its plane is 12mr2\frac{1}{2}mr^221mr2, where mmm is the mass and rrr is the radius.
16. The angular acceleration of a rotating body is defined as?
A) The rate of change of velocity
B) The rate of change of angular velocity
C) The rate of change of angular displacement
D) The rate of change of force
Answer: B) The rate of change of angular velocity.
Explanation: Angular acceleration is the rate at which the angular velocity of a rotating body changes with time, similar to how linear acceleration is the rate of change of linear velocity.
17. What is the relationship between torque τ\tauτ and angular acceleration α\alphaα?
A) τ=Iα\tau = I\alphaτ=Iα
B) τ=mα\tau = m\alphaτ=mα
C) τ=Iα\tau = \frac{I}{\alpha}τ=αI
D) τ=αI\tau = \frac{\alpha}{I}τ=Iα
Answer: A) τ=Iα\tau = I\alphaτ=Iα.
Explanation: Torque τ\tauτ is equal to the moment of inertia III times the angular acceleration α\alphaα, similar to the linear form of Newton’s second law F=maF = maF=ma.
18. The center of mass of a uniform rigid body lies?
A) At the center of the body.
B) At the edge of the body.
C) Outside the body.
D) At any point within the body.
Answer: A) At the center of the body.
Explanation: For a uniform rigid body, the center of mass lies at the geometric center of the body because the mass is evenly distributed.
19. Which of the following is true for a rigid body in equilibrium?
A) The net force and net torque on the body are both zero.
B) The body is stationary.
C) The body moves in a straight line.
D) The body rotates with constant angular velocity.
Answer: A) The net force and net torque on the body are both zero.
Explanation: For a rigid body to be in equilibrium, both the net external force and the net external torque acting on it must be zero, meaning there is no translational or rotational acceleration.
20. What is the rotational equivalent of Newton’s second law?
A) τ=Iα\tau = I\alphaτ=Iα
B) F=maF = maF=ma
C) L=IωL = I\omegaL=Iω
D) T=IrT = \frac{I}{r}T=rI
Answer: A) τ=Iα\tau = I\alphaτ=Iα.
Explanation: The rotational equivalent of Newton’s second law is τ=Iα\tau = I\alphaτ=Iα, where τ\tauτ is torque, III is the moment of inertia, and α\alphaα is angular acceleration.
21. What happens to the moment of inertia if the mass is moved further from the axis of rotation?
A) The moment of inertia decreases.
B) The moment of inertia stays the same.
C) The moment of inertia increases.
D) The moment of inertia becomes zero.
Answer: C) The moment of inertia increases.
Explanation: The moment of inertia increases as mass is moved farther from the axis of rotation, as it depends on both the mass and the square of the distance from the axis.
22. The angular velocity of a rotating body is related to the linear velocity by the equation?
A) v=rωv = r\omegav=rω
B) v=ωrv = \frac{\omega}{r}v=rω
C) v=rαv = r\alphav=rα
D) v=rωv = \frac{r}{\omega}v=ωr
Answer: A) v=rωv = r\omegav=rω.
Explanation: Linear velocity vvv of a point on a rotating object is related to angular velocity ω\omegaω by the equation v=rωv = r\omegav=rω, where rrr is the distance from the axis of rotation.
23. If the moment of inertia of a body is doubled, its rotational kinetic energy will?
A) Increase by a factor of 2.
B) Decrease by a factor of 2.
C) Double.
D) Quadruple.
Answer: A) Increase by a factor of 2.
Explanation: Rotational kinetic energy is given by 12Iω2\frac{1}{2}I\omega^221Iω2. If the moment of inertia III is doubled, the rotational kinetic energy will also double, provided angular velocity remains constant.
24. If the distance from the axis of rotation is halved, the moment of inertia of a particle will?
A) Stay the same.
B) Be halved.
C) Be quadrupled.
D) Be reduced by a factor of 4.
Answer: B) Be halved.
Explanation: Moment of inertia depends on the square of the distance from the axis of rotation. If the distance is halved, the moment of inertia is reduced by a factor of 14\frac{1}{4}41.
25. The angular displacement of a rotating object is?
A) The angle through which an object rotates in a given time.
B) The rate of change of angular velocity.
C) The rate of change of angular acceleration.
D) The total distance traveled by the object in a circular path.
Answer: A) The angle through which an object rotates in a given time.
Explanation: Angular displacement refers to the angle through which a rotating object moves within a certain time period.
26. A rotating object slows down due to?
A) External torque.
B) Internal forces.
C) Friction or resistance.
D) Constant angular velocity.
Answer: C) Friction or resistance.
Explanation: The slowing down of a rotating object is often due to external forces such as friction or air resistance that oppose its motion.
27. The net torque acting on a rigid body is responsible for?
A) Translational motion.
B) Rotational motion.
C) Both translational and rotational motion.
D) Neither translational nor rotational motion.
Answer: B) Rotational motion.
Explanation: The net torque applied to a rigid body causes rotational motion, similar to how a net force causes translational motion.
28. In the case of a uniform circular motion, which quantity remains constant?
A) Speed
B) Velocity
C) Acceleration
D) Force
Answer: A) Speed.
Explanation: In uniform circular motion, the speed (magnitude of velocity) remains constant, but the direction of velocity continuously changes, which causes acceleration.
29. The moment of inertia of a solid sphere about an axis through its center is?
A) 25mr2\frac{2}{5}mr^252mr2
B) 12mr2\frac{1}{2}mr^221mr2
C) 13mr2\frac{1}{3}mr^231mr2
D) mr2mr^2mr2
Answer: A) 25mr2\frac{2}{5}mr^252mr2.
Explanation: The moment of inertia of a solid sphere rotating about an axis through its center is 25mr2\frac{2}{5}mr^252mr2.
30. Which of the following quantities is used to describe the rotational motion of a rigid body?
A) Moment of inertia
B) Angular velocity
C) Torque
D) All of the above
Answer: D) All of the above.
Explanation: Moment of inertia, angular velocity, and torque are all used to describe the rotational motion of a rigid body.
31. The energy required to rotate a body depends on?
A) Its mass
B) Its moment of inertia
C) Its angular velocity
D) All of the above
Answer: D) All of the above.
Explanation: The energy required to rotate a body depends on its mass, moment of inertia, and angular velocity, as the rotational kinetic energy is 12Iω2\frac{1}{2}I\omega^221Iω2.
32. Which of the following is true about the motion of a system of particles under the influence of internal forces only?
A) The system will rotate.
B) The system’s center of mass remains at rest or moves with constant velocity.
C) The system’s center of mass moves in a circle.
D) The system will accelerate.
Answer: B) The system’s center of mass remains at rest or moves with constant velocity.
Explanation: Internal forces do not affect the motion of the center of mass; only external forces do.
33. The total angular momentum of a system of particles is?
A) The sum of the individual angular momenta.
B) Always constant.
C) Zero in a closed system.
D) Dependent on the velocity of particles.
Answer: A) The sum of the individual angular momenta.
Explanation: The total angular momentum of a system is the vector sum of the angular momenta of all individual particles within the system.
34. A rigid body is one in which?
A) The shape of the body changes.
B) The body cannot rotate.
C) The distance between particles in the body remains constant.
D) Only the center of mass moves.
Answer: C) The distance between particles in the body remains constant.
Explanation: A rigid body is one in which the relative positions of particles do not change, meaning the shape and size of the body remain constant.
35. The work done by a torque is related to the change in?
A) Angular velocity.
B) Angular acceleration.
C) Moment of inertia.
D) Rotational kinetic energy.
Answer: D) Rotational kinetic energy.
Explanation: The work done by a torque is related to the change in rotational kinetic energy, similar to how work is related to the change in translational kinetic energy.
36. The rotational inertia of a system is calculated by?
A) I=∑mr2I = \sum mr^2I=∑mr2
B) I=∑mα2I = \sum m\alpha^2I=∑mα2
C) I=∑ω2I = \sum \omega^2I=∑ω2
D) I=∑mI = \sum mI=∑m
Answer: A) I=∑mr2I = \sum mr^2I=∑mr2.
Explanation: The moment of inertia III is calculated by summing the product of the mass of each particle and the square of its distance from the axis of rotation.
37. The work done by a torque is?
A) The rate of change of angular velocity.
B) The rate of change of angular acceleration.
C) Equal to the change in rotational kinetic energy.
D) The force applied times the distance.
Answer: C) Equal to the change in rotational kinetic energy.
Explanation: The work done by a torque on a rotating object is equal to the change in the rotational kinetic energy of the object.
38. Which of the following is true for an object rotating with a constant angular velocity?
A) It has zero angular acceleration.
B) It has constant torque.
C) It is at rest.
D) It has a non-zero moment of inertia.
Answer: A) It has zero angular acceleration.
Explanation: If an object is rotating with constant angular velocity, it has zero angular acceleration, meaning its rotational speed is not changing.
39. The rotational analog of Newton’s first law is?
A) An object will continue rotating at constant angular velocity unless acted upon by an external torque.
B) The net torque on an object is equal to the moment of inertia times the angular acceleration.
C) Torque is equal to force times radius.
D) None of the above.
Answer: A) An object will continue rotating at constant angular velocity unless acted upon by an external torque.
Explanation: This is the rotational equivalent of Newton’s first law, which states that an object will maintain its state of motion unless acted upon by an external force or torque.
40. The torque required to rotate a body is proportional to?
A) The moment of inertia and the angular acceleration.
B) The moment of inertia and the angular velocity.
C) The force and distance.
D) The distance and velocity.
Answer: A) The moment of inertia and the angular acceleration.
Explanation: The torque required to rotate a body is proportional to both its moment of inertia and angular acceleration, as described by τ=Iα\tau = I\alphaτ=Iα.
41. The moment of inertia of a hollow cylinder is?
A) mr2mr^2mr2
B) 12mr2\frac{1}{2}mr^221mr2
C) 12m(r12+r22)\frac{1}{2}m(r_1^2 + r_2^2)21m(r12+r22)
D) 14mr2\frac{1}{4}mr^241mr2
Answer: C) 12m(r12+r22)\frac{1}{2}m(r_1^2 + r_2^2)21m(r12+r22).
Explanation: The moment of inertia of a hollow cylinder is given by 12m(r12+r22)\frac{1}{2}m(r_1^2 + r_2^2)21m(r12+r22), where r1r_1r1 and r2r_2r2 are the inner and outer radii, respectively.
42. A spinning disk slows down due to?
A) Conservation of angular momentum.
B) Frictional forces.
C) Internal forces.
D) Constant angular velocity.
Answer: B) Frictional forces.
Explanation: The spinning disk slows down primarily due to frictional forces, which dissipate energy and reduce the rotational speed.
43. A wheel with a moment of inertia III is rotating with an angular velocity ω\omegaω. What is its angular momentum?
A) IωI\omegaIω
B) 12Iω\frac{1}{2}I\omega21Iω
C) I2ωI^2\omegaI2ω
D) Iω2I \omega^2Iω2
Answer: A) IωI\omegaIω.
Explanation: Angular momentum is given by L=IωL = I\omegaL=Iω, where III is the moment of inertia and ω\omegaω is the angular velocity.
44. The moment of inertia of a ring about an axis through its center perpendicular to the plane of the ring is?
A) 12mr2\frac{1}{2}mr^221mr2
B) mr2mr^2mr2
C) 14mr2\frac{1}{4}mr^241mr2
D) 35mr2\frac{3}{5}mr^253mr2
Answer: B) mr2mr^2mr2.
Explanation: The moment of inertia of a ring about an axis through its center perpendicular to its plane is mr2mr^2mr2, where mmm is the mass and rrr is the radius of the ring.
45. If the angular momentum of a system is constant, then?
A) The system is in equilibrium.
B) The system is rotating with constant angular velocity.
C) There is no external torque acting on the system.
D) The system’s center of mass is at rest.
Answer: C) There is no external torque acting on the system.
Explanation: If the angular momentum of a system is constant, this implies that there is no external torque acting on the system (angular momentum is conserved).
46. The work-energy theorem for rotational motion states that?
A) The work done by torque is equal to the change in angular momentum.
B) The work done by torque is equal to the change in rotational kinetic energy.
C) The work done by torque is equal to the moment of inertia times the change in angular velocity.
D) The work done is equal to the change in angular displacement.
Answer: B) The work done by torque is equal to the change in rotational kinetic energy.
Explanation: The work-energy theorem for rotational motion states that the work done by the torque is equal to the change in rotational kinetic energy of the body.
47. The moment of inertia of a solid cube about an axis through its center is?
A) 16mL2\frac{1}{6}mL^261mL2
B) 12mL2\frac{1}{2}mL^221mL2
C) 13mL2\frac{1}{3}mL^231mL2
D) mL2mL^2mL2
Answer: A) 16mL2\frac{1}{6}mL^261mL2.
Explanation: The moment of inertia of a solid cube about an axis through its center is 16mL2\frac{1}{6}mL^261mL2, where LLL is the side length.
48. In a system of particles, if the total angular momentum is conserved, then?
A) No external torque is acting on the system.
B) The particles are in equilibrium.
C) The velocity of the particles is constant.
D) The system is rotating with constant angular velocity.
Answer: A) No external torque is acting on the system.
Explanation: If the total angular momentum is conserved, it indicates that no external torque is acting on the system, as angular momentum changes only under the influence of torque.
49. In the case of rolling motion, the condition for pure rolling is?
A) The linear velocity is equal to the tangential velocity at the point of contact.
B) The linear velocity is twice the tangential velocity.
C) The object is rotating but not translating.
D) The object is translating but not rotating.
Answer: A) The linear velocity is equal to the tangential velocity at the point of contact.
Explanation: For pure rolling motion, the linear velocity of the center of mass is equal to the tangential velocity of the point of contact with the surface.
50. The unit of angular velocity is?
A) rad/s
B) m/s
C) kg·m/s
D) N·m
Answer: A) rad/s.
Explanation: The unit of angular velocity is radians per second (rad/s), as it measures the rate of change of angular displacement.
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