Parallel and perpendicular axes theorems and applications, equilibrium of rigid bodies - Question Bank

1. The Perpendicular Axes Theorem is applicable to a rigid body if its mass distribution is:
A) Uniform in all directions.
B) Uniform in the plane of the axes.
C) Non-uniform in the plane of the axes.
D) Concentrated at the center.
2. If a rigid body is in equilibrium, then the net work done by all forces during any displacement is:
A) Always positive
B) Always negative
C) Zero
D) Depends on the displacement
3. What is the moment of inertia of a thin rod of mass M and length L about an axis passing through one end and perpendicular to the rod?
A) ML^2 / 12
B) ML^2 / 6
C) ML^2 / 3
D) ML^2 / 2
4. A uniform rod is pivoted at its center. If two equal forces are applied at the ends in opposite directions perpendicular to the rod, the rod is in:
A) Translational equilibrium only.
B) Rotational equilibrium only.
C) Both translational and rotational equilibrium.
D) Neither translational nor rotational equilibrium.
5. For a rigid body to achieve translational equilibrium, the vector sum of all external forces acting on it must be zero. This means:
A) The body's velocity is constant.
B) The body's acceleration is zero.
C) The body's momentum is zero.
D) The body's kinetic energy is zero.
6. Which theorem is essential for calculating the moment of inertia of composite rigid bodies?
A) Parallel Axes Theorem
B) Perpendicular Axes Theorem
C) Both Parallel and Perpendicular Axes Theorems
D) None of the above
7. A rigid body is in equilibrium. If we apply a force at one point and an equal and opposite force at another point, such that they form a couple, what happens?
A) The body will translate.
B) The body will rotate.
C) The body will translate and rotate.
D) The body will remain in equilibrium if the couple is zero.
8. The Parallel Axes Theorem is a consequence of the definition of moment of inertia and:
A) Newton's third law
B) Conservation of angular momentum
C) The definition of the center of mass
D) The concept of torque
9. What is the moment of inertia of a thin uniform rod of mass M and length L about an axis perpendicular to the rod and passing through a point at a distance x from one end?
A) ML^2 / 12 + Mx^2
B) ML^2 / 3 + M(L/2 - x)^2
C) ML^2 / 12 + M(L/2 - x)^2
D) ML^2 / 3 + Mx^2
10. A system is composed of multiple rigid bodies. For the entire system to be in equilibrium:
A) Each individual body must be in equilibrium.
B) The net external force on the system must be zero.
C) The net external torque on the system about any point must be zero.
D) All of the above.
11. For a body to be in equilibrium, the condition Στ = 0 must hold:
A) About the center of mass only.
B) About any point.
C) About the point where the net force acts.
D) About the point where the net torque is maximum.
12. Using the Parallel Axes Theorem, what is the moment of inertia of a uniform circular ring of mass M and radius R about an axis parallel to a diameter and tangent to the ring?
A) MR^2
B) 2MR^2
C) 3MR^2
D) 4MR^2
13. A uniform circular ring of mass M and radius R. What is its moment of inertia about an axis passing through its center and lying in its plane (i.e., a diameter)?
A) MR^2
B) MR^2 / 2
C) 2MR^2
D) 3MR^2 / 2
14. Consider a rigid body in equilibrium. If an additional force is applied such that it creates a net torque but no net force, what happens to the body's motion?
A) It starts to translate with constant acceleration.
B) It starts to rotate with constant angular acceleration.
C) It starts to translate and rotate with constant accelerations.
D) It remains in equilibrium.
15. The Perpendicular Axes Theorem can be applied to:
A) Any rigid body
B) Only planar bodies
C) Only solid bodies
D) Only hollow bodies
16. The Parallel Axes Theorem can be applied to:
A) Any rigid body
B) Only planar bodies
C) Only bodies with uniform mass distribution
D) Only point masses
17. A uniform rod is supported at two points. For the rod to be in equilibrium, the net force on the rod must be zero, and the net torque about any point must be zero. This implies:
A) The rod is at rest.
B) The rod is moving with constant velocity.
C) The rod has constant linear and angular acceleration.
D) The rod has zero linear and angular acceleration.
18. What is the moment of inertia of a thin rod of mass M and length L about an axis passing through a point at a distance L/4 from the center and perpendicular to the rod?
A) ML^2 / 12
B) ML^2 / 3
C) ML^2 / 48
D) 7ML^2 / 48
19. A body is in equilibrium. If we shift the origin of the coordinate system, will the conditions for equilibrium change?
A) Yes, the net force will change.
B) Yes, the net torque will change.
C) No, as long as the body is in equilibrium, the net force and net torque will remain zero.
D) It depends on the direction of shift.
20. Using the Perpendicular Axes Theorem, what is the moment of inertia of a uniform rectangular plate of mass M and dimensions a x b about an axis passing through its center and perpendicular to its plane?
A) M(a^2 + b^2) / 12
B) Mb^2 / 12
C) Ma^2 / 12
D) M(a^2 + b^2) / 6
21. What is the moment of inertia of a uniform rectangular plate of mass M and dimensions a x b about an axis passing through its center and parallel to side 'b'?
A) M(a^2 + b^2) / 12
B) Mb^2 / 12
C) Ma^2 / 12
D) M(a^2 + b^2) / 6
22. A uniform rectangular plate of mass M and dimensions a x b. What is its moment of inertia about an axis passing through its center and parallel to side 'a'?
A) M(a^2 + b^2) / 12
B) Mb^2 / 12
C) Ma^2 / 12
D) M(a^2 + b^2) / 6
23. Consider a system of two point masses m1 and m2 separated by a distance r. If this system is in equilibrium, what must be true?
A) Both masses must be at rest.
B) The net force on the system is zero.
C) The net torque on the system about any point is zero.
D) Both B and C must be true.
24. The Perpendicular Axes Theorem is derived from the principle of:
A) Isotropy of mass distribution
B) Anisotropy of mass distribution
C) Conservation of energy
D) Conservation of momentum
25. Using the Parallel Axes Theorem, what is the moment of inertia of a thin hollow cylinder of mass M and radius R about an axis parallel to its central axis and at a distance R from it?
A) MR^2
B) 2MR^2
C) 3MR^2
D) 4MR^2
26. What is the moment of inertia of a thin hollow cylinder of mass M and radius R about its central axis?
A) MR^2 / 2
B) MR^2
C) 2MR^2 / 3
D) MR^2 / 4
27. A uniform rod of mass M and length L is pivoted at one end. If a force F is applied at the other end perpendicular to the rod, what is the torque about the pivot?
A) FL
B) FL/2
C) 0
D) 2FL
28. If a body is in translational equilibrium, its linear momentum is constant. If it is also in rotational equilibrium, its angular momentum is:
A) Zero
B) Constant
C) Increasing
D) Decreasing
29. The Parallel Axes Theorem is used to find the moment of inertia about an axis that is:
A) Passing through the center of mass.
B) Perpendicular to the plane of a planar body.
C) Parallel to an axis passing through the center of mass.
D) Passing through any point in the body.
30. Consider a rigid body acted upon by forces F1, F2, ... Fn and torques τ1, τ2, ... τn. For equilibrium, what must be true?
A) ΣFi = 0
B) Στi = 0
C) Both ΣFi = 0 and Στi = 0
D) ΣFi = 0 or Στi = 0
31. Which of the following is NOT a condition for the equilibrium of a rigid body?
A) Sum of all external forces is zero.
B) Sum of all external torques about any point is zero.
C) The body must be at rest.
D) The center of mass of the body has zero acceleration.
32. A ladder is leaning against a wall. For the ladder to be in equilibrium, which conditions must be met?
A) Net force is zero only.
B) Net torque about any point is zero only.
C) Net force and net torque about any point are zero.
D) The ladder must be vertical.
33. For a rigid body to be in equilibrium, the sum of all torques acting on it about any point must be zero. This is the condition for:
A) Rotational equilibrium
B) Translational equilibrium
C) Both translational and rotational equilibrium
D) Neither translational nor rotational equilibrium
34. For a rigid body to be in equilibrium, the sum of all forces acting on it must be zero. This is the condition for:
A) Rotational equilibrium
B) Translational equilibrium
C) Both translational and rotational equilibrium
D) Neither translational nor rotational equilibrium
35. A uniform solid sphere of mass M and radius R has a moment of inertia I_cm = (2/5)MR^2 about an axis through its center. What is its moment of inertia about a parallel axis tangent to its surface?
A) (2/5)MR^2
B) (7/5)MR^2
C) (3/5)MR^2
D) MR^2
36. Apply the Perpendicular Axes Theorem to find the moment of inertia of a uniform disc about a diameter. Given I_z = MR^2/2 for an axis perpendicular to the plane through the center. Since the disc is uniform and symmetric, I_x = I_y for any diameter.
A) MR^2 / 2
B) MR^2 / 4
C) MR^2
D) 2MR^2
37. What is the moment of inertia of a uniform disc of mass M and radius R about a diameter?
A) MR^2
B) MR^2 / 2
C) MR^2 / 4
D) 2MR^2
38. Consider a uniform disc of mass M and radius R. What is its moment of inertia about an axis passing through its center and perpendicular to its plane?
A) MR^2
B) MR^2 / 2
C) 2MR^2 / 5
D) 3MR^2 / 2
39. A rigid body is in equilibrium. Which of the following statements is always true?
A) It is at rest.
B) It is moving with constant velocity.
C) It has constant linear and angular velocity.
D) It has zero net force and zero net torque acting on it.
40. Rotational equilibrium for a rigid body means that the net external torque acting on it about any point is zero. What does this imply?
A) The body's angular acceleration is zero.
B) The body's center of mass has zero acceleration.
C) The body's angular momentum is constant.
D) Both A and C
41. Translational equilibrium for a rigid body means that the net external force acting on it is zero. What does this imply?
A) The body's center of mass has zero acceleration.
B) The body's angular acceleration is zero.
C) The body's linear momentum is constant.
D) All of the above
42. What is the condition for a rigid body to be in equilibrium?
A) Net force is zero and net torque is zero.
B) Net force is zero or net torque is zero.
C) Net force is non-zero and net torque is zero.
D) Net force is zero and net torque is non-zero.
43. Using the Parallel Axes Theorem, what is the moment of inertia of a thin uniform rod of mass M and length L about an axis passing through one of its ends and perpendicular to its length?
A) ML^2 / 12
B) ML^2 / 3
C) ML^2 / 2
D) ML^2
44. Consider a thin uniform rod of mass M and length L. What is its moment of inertia about an axis passing through its center and perpendicular to its length?
A) ML^2 / 12
B) ML^2 / 3
C) ML^2 / 2
D) ML^2
45. For a planar body, if I_x and I_y are the moments of inertia about two perpendicular axes in the plane, and I_z is the moment of inertia about the axis perpendicular to the plane passing through the intersection of x and y axes, what is the relation according to the Perpendicular Axes Theorem?
A) I_z = I_x + I_y
B) I_z = I_x - I_y
C) I_z = I_x * I_y
D) I_z = I_x / I_y
46. The Perpendicular Axes Theorem is applicable to planar bodies. What does it state?
A) The sum of moments of inertia about two perpendicular axes in the plane of the body is equal to the moment of inertia about the axis perpendicular to the plane and passing through their intersection.
B) The difference of moments of inertia about two perpendicular axes in the plane of the body is equal to the moment of inertia about the axis perpendicular to the plane and passing through their intersection.
C) The product of moments of inertia about two perpendicular axes in the plane of the body is equal to the moment of inertia about the axis perpendicular to the plane and passing through their intersection.
D) The average of moments of inertia about two perpendicular axes in the plane of the body is equal to the moment of inertia about the axis perpendicular to the plane and passing through their intersection.
47. In the Parallel Axes Theorem, 'd' represents the perpendicular distance between the two parallel axes. What is the unit of 'd'?
A) Meters (m)
B) Kilograms (kg)
C) Seconds (s)
D) Newtons (N)
48. In the Parallel Axes Theorem, 'I_cm' represents the moment of inertia about an axis passing through the center of mass. What does 'M' represent?
A) Mass of the body
B) Moment of inertia
C) Distance from the center of mass
D) Angular velocity
49. The Parallel Axes Theorem relates the moment of inertia about an axis passing through the center of mass to the moment of inertia about a parallel axis. What is the relationship?
A) I = I_cm - Md^2
B) I = I_cm + Md^2
C) I = Md^2 - I_cm
D) I = I_cm / Md^2
50. Which theorem allows us to calculate the moment of inertia of a body about an axis parallel to an axis passing through its center of mass?
A) Perpendicular Axes Theorem
B) Parallel Axes Theorem
C) Theorem of Superposition
D) Conservation of Angular Momentum Theorem
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