Parallel and perpendicular axes theorems and applications, equilibrium of rigid bodies - One Line Questions

1. 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? — (7/5)MR^2
2. For a body to be in equilibrium, the condition Στ = 0 must hold: — About any point.
3. If a rigid body is in equilibrium, then the net work done by all forces during any displacement is: — Zero
4. The Parallel Axes Theorem can be applied to: — Any rigid body
5. The Perpendicular Axes Theorem can be applied to: — Only planar bodies
6. 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? — Both B and C must be true.
7. A system is composed of multiple rigid bodies. For the entire system to be in equilibrium: — All of the above.
8. 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? — FL
9. 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? — I = I_cm + Md^2
10. 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? — I_z = I_x + I_y
11. The Perpendicular Axes Theorem is derived from the principle of: — Isotropy of mass distribution
12. A rigid body is in equilibrium. Which of the following statements is always true? — It has zero net force and zero net torque acting on it.
13. 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? — It starts to rotate with constant angular acceleration.
14. 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'? — Mb^2 / 12
15. 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'? — Ma^2 / 12
16. 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? — M(a^2 + b^2) / 12
17. 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? — Mass of the body
18. In the Parallel Axes Theorem, 'd' represents the perpendicular distance between the two parallel axes. What is the unit of 'd'? — Meters (m)
19. 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? — ML^2 / 12
20. 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? — ML^2 / 3
21. 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? — 7ML^2 / 48
22. 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? — ML^2 / 3
23. 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? — ML^2 / 12 + M(L/2 - x)^2
24. 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? — MR^2 / 2
25. What is the moment of inertia of a uniform disc of mass M and radius R about a diameter? — MR^2 / 4
26. 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? — 3MR^2
27. 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)? — MR^2
28. 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? — 3MR^2
29. 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. — MR^2 / 4
30. What is the moment of inertia of a thin hollow cylinder of mass M and radius R about its central axis? — MR^2
31. What is the condition for a rigid body to be in equilibrium? — Net force is zero and net torque is zero.
32. A ladder is leaning against a wall. For the ladder to be in equilibrium, which conditions must be met? — Net force and net torque about any point are zero.
33. The Parallel Axes Theorem is a consequence of the definition of moment of inertia and: — The definition of the center of mass
34. Which theorem is essential for calculating the moment of inertia of composite rigid bodies? — Both Parallel and Perpendicular Axes Theorems
35. The Parallel Axes Theorem is used to find the moment of inertia about an axis that is: — Parallel to an axis passing through the center of mass.
36. 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? — Parallel Axes Theorem
37. For a rigid body to be in equilibrium, the sum of all forces acting on it must be zero. This is the condition for: — Translational equilibrium
38. 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: — Rotational equilibrium
39. Which of the following is NOT a condition for the equilibrium of a rigid body? — The body must be at rest.
40. 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? — The body will rotate.
41. Rotational equilibrium for a rigid body means that the net external torque acting on it about any point is zero. What does this imply? — Both A and C
42. Translational equilibrium for a rigid body means that the net external force acting on it is zero. What does this imply? — All of the above
43. For a rigid body to achieve translational equilibrium, the vector sum of all external forces acting on it must be zero. This means: — The body's acceleration is zero.
44. 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: — The rod has zero linear and angular acceleration.
45. The Perpendicular Axes Theorem is applicable to planar bodies. What does it state? — 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.
46. 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: — Both translational and rotational equilibrium.
47. The Perpendicular Axes Theorem is applicable to a rigid body if its mass distribution is: — Uniform in the plane of the axes.
48. A body is in equilibrium. If we shift the origin of the coordinate system, will the conditions for equilibrium change? — No, as long as the body is in equilibrium, the net force and net torque will remain zero.
49. If a body is in translational equilibrium, its linear momentum is constant. If it is also in rotational equilibrium, its angular momentum is: — Constant
50. Consider a rigid body acted upon by forces F1, F2, ... Fn and torques τ1, τ2, ... τn. For equilibrium, what must be true? — Both ΣFi = 0 and Στi = 0
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