Centre of mass of particle systems and rigid bodies, basic concepts of rotational motion - Question Bank

1. Which of the following represents the angular equivalent of Newton's second law of motion (F=ma)?
A) τ = Iω
B) τ = Iα
C) L = Iω
D) E = 1/2 Iω^2
2. If a force F acts at a position vector r from the origin, the torque τ is given by:
A) τ = r . F
B) τ = r x F
C) τ = F x r
D) τ = r / F
3. The moment of inertia depends on the mass distribution relative to the:
A) Center of the body
B) Geometric center
C) Axis of rotation
D) Surface of the body
4. Which quantity is conserved for a rigid body rotating about a fixed axis in the absence of external torque?
A) Linear momentum
B) Angular momentum
C) Kinetic energy
D) Potential energy
5. When a rigid body rotates about a fixed axis, all points on the body have the same:
A) Linear velocity
B) Linear acceleration
C) Angular velocity
D) Tangential acceleration
6. The angular acceleration of a body is zero. This implies that:
A) The body is at rest.
B) The body is rotating with constant angular velocity.
C) The body is not rotating.
D) The net torque on the body is zero.
7. Which of the following is NOT a unit of angular velocity?
A) radians per second
B) degrees per second
C) revolutions per minute (RPM)
D) Hertz (Hz)
8. For a continuous rigid body, the center of mass can be found using integration: X_cm = (1/M) * integral(x dm). Here, dm represents:
A) The total mass of the body.
B) An infinitesimal element of mass.
C) The average mass of the body.
D) The mass of the center of mass.
9. What is the center of mass of a system of two particles of mass m1 and m2, separated by distance r?
A) It divides the distance in the ratio m1:m2.
B) It divides the distance in the ratio m2:m1.
C) It is always at the midpoint.
D) It depends on their velocities.
10. If a body is subjected to a net external torque, its angular momentum will change, resulting in:
A) Constant angular velocity
B) Angular acceleration
C) Constant linear velocity
D) Linear acceleration
11. A body is in equilibrium if the net external force and the net external torque acting on it are both:
A) Maximum
B) Minimum
C) Zero
D) Equal
12. For a system of particles, if the net external torque is zero, then the total angular momentum of the system:
A) Increases
B) Decreases
C) Remains constant
D) Becomes zero
13. A dancer spins faster when she pulls her arms inwards. This is an example of the conservation of:
A) Linear momentum
B) Angular momentum
C) Energy
D) Mass
14. If the angular velocity of a rotating body is doubled, its rotational kinetic energy becomes:
A) Half
B) Double
C) Four times
D) One-fourth
15. Which of the following has the largest moment of inertia when rotating about an axis through its center?
A) A thin rod of length L
B) A solid sphere of radius R
C) A thin circular disc of radius R
D) A hollow sphere of radius R
16. Rotational kinetic energy of a body with moment of inertia I and angular velocity ω is:
A) 1/2 Iω
B) 1/2 Iω^2
C) Iω^2
D) 1/2 ωI^2
17. The work done by torque (τ) during an angular displacement (dθ) is:
A) τ dθ
B) τ / dθ
C) dθ / τ
D) τ + dθ
18. If a body is undergoing pure rotation, the acceleration of a particle at a distance r from the axis is:
A) v^2/r (centripetal)
B) rα (tangential)
C) Both v^2/r and rα
D) Zero
19. The angular velocity vector (ω) is directed along the axis of rotation according to the:
A) Right-hand rule.
B) Left-hand rule.
C) Newton's first law.
D) Ampere's law.
20. What happens to the center of mass of a rigid body when it rotates?
A) It always moves.
B) It may or may not move, depending on the axis of rotation.
C) It always remains stationary.
D) It moves in a circular path.
21. If the net external force on a system of particles is zero, then the velocity of the center of mass:
A) Is always zero.
B) Is always maximum.
C) Remains constant.
D) Is always undefined.
22. Consider a system of two identical particles. If one particle is at the origin and the other is at (1m, 0), the center of mass is at:
A) (0, 0)
B) (0.5m, 0)
C) (1m, 0)
D) (0.25m, 0)
23. Which of the following statements is correct regarding the center of mass of a system?
A) It always lies inside the system.
B) It is the point where the weight of the system acts.
C) It can lie outside the system.
D) It is independent of the frame of reference.
24. The radius of gyration (k) of a rigid body is defined such that its moment of inertia (I) is equal to:
A) Mk
B) Mk^2
C) M/k
D) M/k^2
25. The moment of inertia of a hollow sphere of radius R and mass M about an axis passing through its center is:
A) MR^2
B) 2/5 MR^2
C) 2/3 MR^2
D) 2 MR^2/3
26. The moment of inertia of a solid sphere of radius R and mass M about an axis passing through its center is:
A) MR^2
B) 2/5 MR^2
C) 2/3 MR^2
D) 2/3 MR^2
27. The moment of inertia of a thin uniform circular disc of radius R and mass M about an axis passing through its center and perpendicular to its plane is:
A) MR^2
B) 1/2 MR^2
C) 2/5 MR^2
D) MR^2/2
28. A thin uniform rod of length L and mass M, rotating about an axis passing through one of its ends and perpendicular to its length, has a moment of inertia of:
A) ML^2/12
B) ML^2/3
C) ML^2/2
D) ML^2
29. A thin uniform rod of length L and mass M, rotating about an axis passing through its center and perpendicular to its length, has a moment of inertia of:
A) ML^2/12
B) ML^2/3
C) ML^2/2
D) ML^2
30. In the absence of an external torque, the total angular momentum of a system remains conserved. This is the principle of:
A) Conservation of Linear Momentum
B) Conservation of Energy
C) Conservation of Angular Momentum
D) Conservation of Mass
31. The rate of change of angular momentum of a particle is equal to the:
A) Net external force acting on it.
B) Net external torque acting on it.
C) Net internal force acting on it.
D) Net internal torque acting on it.
32. Angular momentum (L) of a particle is defined as the product of its position vector (r) and its:
A) Linear velocity (v)
B) Linear momentum (p)
C) Angular velocity (ω)
D) Angular acceleration (α)
33. The Parallel Axis Theorem states that I = I_cm + Md^2, where I is the moment of inertia about an axis, I_cm is the moment of inertia about a parallel axis through the center of mass, M is the total mass, and d is the:
A) Distance between the two axes.
B) Radius of gyration.
C) Angular velocity.
D) Linear velocity.
34. Which theorem relates the moment of inertia about an axis through the center of mass to the moment of inertia about a parallel axis?
A) Work-Energy Theorem
B) Parallel Axis Theorem
C) Perpendicular Axis Theorem
D) Conservation of Angular Momentum Theorem
35. For a system of particles, the moment of inertia is the sum of the moments of inertia of individual particles about the same axis. This is:
A) Incorrect, it's the sum of masses.
B) Incorrect, it depends on the distance of the center of mass.
C) Correct, I = sum(mi*ri^2).
D) Correct, I = sum(mi*ri).
36. The moment of inertia of a point mass m at a distance r from the axis of rotation is:
A) mr
B) m/r
C) mr^2
D) m^2r
37. Moment of inertia is a measure of an object's resistance to changes in its:
A) Linear motion.
B) Rotational motion.
C) Vibrational motion.
D) Translational motion.
38. If a body is rotating with constant angular velocity, what is its angular acceleration?
A) Maximum
B) Minimum
C) Zero
D) Infinite
39. For a rigid body rotating about a fixed axis, the net external torque equals the product of:
A) Moment of inertia and angular velocity.
B) Moment of inertia and angular acceleration.
C) Mass and angular acceleration.
D) Moment of inertia and linear acceleration.
40. What is the unit of torque?
A) Joule (J)
B) Newton (N)
C) Newton-meter (N-m)
D) Pascal (Pa)
41. Torque is the rotational analog of:
A) Force
B) Momentum
C) Energy
D) Mass
42. The relationship between linear velocity (v) and angular velocity (ω) for a point at a distance r from the axis of rotation is:
A) v = ωr
B) v = ω/r
C) v = r/ω
D) v = ω + r
43. Angular acceleration is the rate of change of:
A) Angular displacement with respect to time.
B) Angular velocity with respect to time.
C) Linear velocity with respect to time.
D) Linear displacement with respect to time.
44. What is the unit of angular velocity?
A) meters per second (m/s)
B) radians per second (rad/s)
C) radians per second squared (rad/s^2)
D) newton-meter (N-m)
45. For a discrete system of n particles, the coordinates of the center of mass (X_cm, Y_cm, Z_cm) are given by:
A) X_cm = sum(mi*xi)/sum(mi), Y_cm = sum(mi*yi)/sum(mi), Z_cm = sum(mi*zi)/sum(mi)
B) X_cm = sum(xi)/n, Y_cm = sum(yi)/n, Z_cm = sum(zi)/n
C) X_cm = sum(mi*xi)/n, Y_cm = sum(mi*yi)/n, Z_cm = sum(mi*zi)/n
D) X_cm = sum(xi)/sum(mi), Y_cm = sum(yi)/sum(mi), Z_cm = sum(zi)/sum(mi)
46. Consider a system of two particles of equal mass placed at (x1, y1) and (x2, y2). The x-coordinate of the center of mass is:
A) (x1 + x2) / 2
B) x1
C) x2
D) (x1 * x2) / (x1 + x2)
47. What is the condition for the center of mass of a system of particles to remain at rest or move with constant velocity?
A) The net external force on the system must be zero.
B) The net external torque on the system must be zero.
C) The net internal force must be zero.
D) The system must be rotating with constant angular velocity.
48. If a rigid body is homogeneous and has uniform density, where is its center of mass located?
A) At its geometric center.
B) At the point of highest density.
C) At the point of lowest density.
D) It can be anywhere within the body.
49. For a system of two particles with masses m1 and m2 and position vectors r1 and r2, the position vector of the center of mass (R_cm) is given by:
A) R_cm = (m1*r1 + m2*r2) / (m1 + m2)
B) R_cm = (m1*r2 + m2*r1) / (m1 + m2)
C) R_cm = (r1 + r2) / 2
D) R_cm = sqrt(m1*r1^2 + m2*r2^2) / (m1 + m2)
50. What is the definition of the center of mass for a system of particles?
A) The point where all forces act on the system.
B) The point representing the average position of all the mass in the system.
C) The point where the system is perfectly balanced horizontally.
D) The geometric center of the system, irrespective of mass distribution.
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