Central Orbits and Moment of Inertia - Question Bank
1. Consider a particle moving under a central force. If the force is repulsive and inversely proportional to the square of the distance, the orbit for a positive energy system is:
2. The conservation of angular momentum in a central orbit implies that the force is always directed along the line connecting the particle and the fixed point. This is true for which type of force?
3. What is the moment of inertia of a particle of mass m, moving in a circle of radius r with speed v, about the center of the circle?
4. For a stable circular orbit under a central force F(r), the effective potential V_eff(r) = L²/(2mr²) + V(r) must have a minimum at the orbital radius. What is the condition for this minimum?
5. In the differential equation for a central orbit, d²u/dθ² + u = F(1/u) / (m h² u²), where u = 1/r, what does F(1/u) represent?
6. What is the moment of inertia of a diatomic molecule (like O2) about its center of mass, assuming the two atoms are point masses m1 and m2 separated by distance r?
7. The radial velocity of a particle in a central orbit is zero when the particle is at:
8. If the total energy of a particle in a central force field is positive, the orbit is:
9. If the total energy of a particle in a central force field is zero, the orbit is:
10. A particle is moving in a central force field. If its total energy is negative, the orbit is:
11. Which of the following shapes has the smallest moment of inertia for the same mass and radius, about an axis through its center?
12. The moment of inertia of a body depends on which of the following factors?
13. If the central force is F(r) = -k r, what is the shape of the orbit?
14. Consider a particle of mass m moving with speed v at a distance r from a fixed point, with velocity perpendicular to the position vector. Its angular momentum is given by:
15. What is the moment of inertia of a thin hoop of mass M and radius R about a diameter?
16. What is the moment of inertia of a thin hoop of mass M and radius R about an axis passing through its center and perpendicular to its plane?
17. The period 'T' of a planet in an elliptical orbit around the Sun is related to the semi-major axis 'a' by Kepler's Third Law. What is the proportionality?
18. If the angular momentum of a particle in a central orbit is zero, what does this imply about its motion?
19. What is the moment of inertia of a uniform rectangular plate of mass M, length L, and width W about an axis passing through its center and perpendicular to its plane?
20. For a central force F(r) = -k/r^n, what is the condition for the orbit to be a closed ellipse (for bound orbits)?
21. Consider an elliptical orbit with semi-major axis 'a' and semi-minor axis 'b'. What is the constant angular momentum per unit mass 'h' related to the area swept per unit time?
22. If a particle's trajectory is a circle of radius r, and the central force F is always directed towards the center, the orbit equation is satisfied if:
23. The torque (τ) acting on a rigid body is related to its moment of inertia (I) and angular acceleration (α) by which equation?
24. Which physical quantity is the rotational analogue of mass?
25. What is the relationship between rotational kinetic energy (K_rot) and moment of inertia (I) and angular velocity (ω)?
26. In the Perpendicular Axis Theorem, I_x and I_y are moments of inertia about axes in the plane of the object, and I_z is about an axis perpendicular to the plane. The theorem holds for objects in which plane?
27. The Perpendicular Axis Theorem applies to planar objects. What is the relationship it states?
28. In the Parallel Axis Theorem, what does 'd' represent?
29. The Parallel Axis Theorem relates the moment of inertia about an axis through the center of mass to the moment of inertia about a parallel axis. What is the formula?
30. What is the moment of inertia of a thin spherical shell of mass M and radius R about an axis passing through its center?
31. What is the moment of inertia of a hollow cylinder (thin shell) of mass M and radius R about its central axis?
32. What is the moment of inertia of a thin circular disk of mass M and radius R about an axis passing through its center and perpendicular to its plane?
33. What is the moment of inertia of a uniform solid sphere of mass M and radius R about an axis passing through its center?
34. What is the moment of inertia of a uniform solid cylinder of mass M and radius R about its central axis?
35. What is the moment of inertia of a thin rod of mass M and length L about an axis passing through one of its ends and perpendicular to its length?
36. What is the moment of inertia of a thin rod of mass M and length L about an axis passing through its center and perpendicular to its length?
37. What is the moment of inertia of a point mass 'm' rotating at a distance 'r' from the axis of rotation?
38. For a continuous rigid body, the moment of inertia is calculated using an integral. What is the general formula?
39. How is the moment of inertia of a system of discrete particles calculated?
40. What is the definition of Moment of Inertia (I)?
41. For a particle in a central orbit, if the angular momentum 'L' is constant, what is the relationship between L, m (mass), r (distance), and v (speed)?
42. The line joining the particle to the center of force sweeps out equal areas in equal times. This is a statement of which law?
43. What is the term 'apsidal distance' in a central orbit?
44. For a particle moving in a circular orbit under a central force, the force must be directed towards the center and have a magnitude equal to:
45. If the central force is attractive and inversely proportional to the square of the distance (F(r) = -k/r²), what is the shape of the orbit for a bound system?
46. What does 'h' represent in the context of a central orbit differential equation?
47. The differential equation describing a central orbit is often expressed in polar coordinates (r, θ). What is the form of this equation?
48. Which physical quantity is conserved for a particle moving under a central force?
49. In the context of central orbits, what is the defining characteristic of a central force?