Central Orbits and Moment of Inertia - One Line Questions

1. 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? — (m1*m2)r²/ (m1+m2)
2. 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? —
3. Which physical quantity is the rotational analogue of mass? — Moment of inertia
4. 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? — Central force
5. The differential equation describing a central orbit is often expressed in polar coordinates (r, θ). What is the form of this equation? — d²r/dθ² + r = F(r) / (m * h² / r³)
6. 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? — dV_eff/dr = 0 and d²V_eff/dr² > 0
7. 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: — Hyperbola
8. A particle is moving in a central force field. If its total energy is negative, the orbit is: — Elliptical
9. If the total energy of a particle in a central force field is zero, the orbit is: — Parabolic
10. If the total energy of a particle in a central force field is positive, the orbit is: — Hyperbolic
11. 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: — F = m * h² / r³
12. 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? — h = 2 * Area / time
13. For a continuous rigid body, the moment of inertia is calculated using an integral. What is the general formula? — I = ∫ r² dm
14. 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? — I = I_cm + Md²
15. The Perpendicular Axis Theorem applies to planar objects. What is the relationship it states? — I_z = I_x + I_y
16. In the context of central orbits, what is the defining characteristic of a central force? — It is always directed towards or away from a fixed point in space.
17. What is the relationship between rotational kinetic energy (K_rot) and moment of inertia (I) and angular velocity (ω)? — K_rot = 1/2 I ω²
18. The line joining the particle to the center of force sweeps out equal areas in equal times. This is a statement of which law? — Kepler's Second Law of Planetary Motion
19. 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)? — L = mvr sin(φ)
20. Which physical quantity is conserved for a particle moving under a central force? — Angular momentum
21. 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: — m * v² / r
22. 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? — M(L² + W²)/12
23. The moment of inertia of a body depends on which of the following factors? — Mass distribution and axis of rotation
24. 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? — ML²/12
25. 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? — ML²/3
26. What is the moment of inertia of a point mass 'm' rotating at a distance 'r' from the axis of rotation? — mr²
27. What is the moment of inertia of a uniform solid sphere of mass M and radius R about an axis passing through its center? — 2/5 MR²
28. What is the moment of inertia of a thin spherical shell of mass M and radius R about an axis passing through its center? — 2/3 MR²
29. What is the moment of inertia of a uniform solid cylinder of mass M and radius R about its central axis? — MR²/2
30. 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? — MR²/2
31. What is the moment of inertia of a hollow cylinder (thin shell) of mass M and radius R about its central axis? — MR²
32. 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? — MR²
33. What is the moment of inertia of a thin hoop of mass M and radius R about a diameter? — MR²/2
34. 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? — mv²
35. 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: — mvr
36. For a central force F(r) = -k/r^n, what is the condition for the orbit to be a closed ellipse (for bound orbits)? — n = 2
37. If the central force is F(r) = -k r, what is the shape of the orbit? — Ellipse or Circle
38. Which of the following shapes has the smallest moment of inertia for the same mass and radius, about an axis through its center? — Solid sphere
39. How is the moment of inertia of a system of discrete particles calculated? — Sum of (mass * distance from axis)²
40. 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? — T ∝ a^(3/2)
41. In the Parallel Axis Theorem, what does 'd' represent? — The distance from the center of mass to the parallel axis.
42. 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? — The force at distance r
43. What does 'h' represent in the context of a central orbit differential equation? — The magnitude of the angular momentum per unit mass
44. What is the term 'apsidal distance' in a central orbit? — The maximum distance from the center of force.
45. If the angular momentum of a particle in a central orbit is zero, what does this imply about its motion? — The particle is moving in a straight line through the center of force.
46. The radial velocity of a particle in a central orbit is zero when the particle is at: — Both periapsis and apoapsis.
47. What is the definition of Moment of Inertia (I)? — The tendency of an object to resist changes in its rotational motion.
48. 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? — XY plane
49. The torque (τ) acting on a rigid body is related to its moment of inertia (I) and angular acceleration (α) by which equation? — τ = Iα
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