TRB BT Assistant - Central Orbits and Moment of Inertia Online Test (MCQ Quiz)

1. In the context of central orbits, what is the defining characteristic of a central force?

A) It is always perpendicular to the velocity vector. B) It is always directed towards or away from a fixed point in space. C) It is proportional to the square of the distance from the fixed point. D) It is independent of the position of the particle.

2. Which physical quantity is conserved for a particle moving under a central force?

A) Linear momentum B) Kinetic energy C) Angular momentum D) Total mechanical energy

3. The differential equation describing a central orbit is often expressed in polar coordinates (r, θ). What is the form of this equation?

A) d²r/dθ² + r = F(r) / (m * h² / r³) B) d²r/dθ² + r = F(r) / (m * h²) C) d²r/dθ² - r = F(r) / (m * h² / r³) D) d²r/dθ² + r = F(r) / (m * r)

4. What does 'h' represent in the context of a central orbit differential equation?

A) The mass of the particle B) The magnitude of the angular momentum per unit mass C) The distance from the fixed center D) The force constant

5. 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?

A) A straight line B) A circle C) An ellipse D) A parabola

6. 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:

A) m * v² / r B) m * v * r C) m * v / r² D) m * r / v²

7. What is the term 'apsidal distance' in a central orbit?

A) The maximum distance from the center of force. B) The minimum distance from the center of force. C) The distance at which the velocity is maximum. D) The average distance from the center of force.

8. The line joining the particle to the center of force sweeps out equal areas in equal times. This is a statement of which law?

A) Kepler's First Law of Planetary Motion B) Kepler's Second Law of Planetary Motion C) Kepler's Third Law of Planetary Motion D) Newton's Law of Universal Gravitation

9. 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)?

A) L = mvr B) L = mvr sin(φ) C) L = m r² ω D) L = m v / r

10. What is the definition of Moment of Inertia (I)?

A) The resistance of an object to linear acceleration. B) The tendency of an object to resist changes in its rotational motion. C) The product of mass and velocity. D) The sum of kinetic and potential energy.

11. How is the moment of inertia of a system of discrete particles calculated?

A) Sum of (mass * distance from axis) B) Sum of (mass * distance from axis)² C) Sum of (mass / distance from axis)² D) Sum of (mass / distance from axis)

12. For a continuous rigid body, the moment of inertia is calculated using an integral. What is the general formula?

A) I = ∫ r² dm B) I = ∫ r dm C) I = ∫ dm / r² D) I = ∫ dm * r

13. What is the moment of inertia of a point mass 'm' rotating at a distance 'r' from the axis of rotation?

A) mr B) mr² C) m/r D) m²/r²

14. 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?

A) ML²/2 B) ML²/3 C) ML²/4 D) ML²/12

15. 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?

A) ML²/2 B) ML²/3 C) ML²/4 D) ML²/12

16. What is the moment of inertia of a uniform solid cylinder of mass M and radius R about its central axis?

A) MR²/2 B) MR² C) 2/5 MR² D) 2/3 MR²

17. What is the moment of inertia of a uniform solid sphere of mass M and radius R about an axis passing through its center?

A) MR² B) 2/3 MR² C) 2/5 MR² D) 1/2 MR²

18. 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?

A) MR²/2 B) MR² C) 2/5 MR² D) ML²/2

19. What is the moment of inertia of a hollow cylinder (thin shell) of mass M and radius R about its central axis?

A) MR²/2 B) MR² C) 2/3 MR² D) 1/2 MR²

20. What is the moment of inertia of a thin spherical shell of mass M and radius R about an axis passing through its center?

A) MR² B) 2/3 MR² C) 2/5 MR² D) 3/2 MR²

21. 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?

A) I = I_cm + Md² B) I = I_cm - Md² C) I = I_cm * Md² D) I = I_cm / Md²

22. In the Parallel Axis Theorem, what does 'd' represent?

A) The distance from the center of mass to the parallel axis. B) The diameter of the object. C) The length of the object. D) The distance from the axis to the farthest point.

23. The Perpendicular Axis Theorem applies to planar objects. What is the relationship it states?

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

24. 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?

A) XY plane B) YZ plane C) XZ plane D) Any plane

25. What is the relationship between rotational kinetic energy (K_rot) and moment of inertia (I) and angular velocity (ω)?

A) K_rot = 1/2 I ω B) K_rot = I ω² C) K_rot = 1/2 I ω² D) K_rot = I² ω

26. Which physical quantity is the rotational analogue of mass?

A) Angular velocity B) Torque C) Moment of inertia D) Angular momentum

27. The torque (τ) acting on a rigid body is related to its moment of inertia (I) and angular acceleration (α) by which equation?

A) τ = Iα B) τ = I/α C) τ = α/I D) τ = I + α

28. 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:

A) F = m * h² / r³ B) F = m * h² / r C) F = m * h / r³ D) F = m * h² / r²

29. 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?

A) h = Area / time B) h = 2 * Area / time C) h = Area / (2 * time) D) h = time / Area

30. For a central force F(r) = -k/r^n, what is the condition for the orbit to be a closed ellipse (for bound orbits)?

A) n = 1 B) n = 2 C) n = 3 D) n = 0

31. 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?

A) M(L² + W²)/12 B) M(L² + W²)/4 C) M(L² + W²)/2 D) M(L² - W²)/12

32. If the angular momentum of a particle in a central orbit is zero, what does this imply about its motion?

A) The particle is moving in a circle. B) The particle is moving in a straight line through the center of force. C) The particle is at rest. D) The particle is moving in an ellipse.

33. 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?

A) T ∝ a B) T ∝ a² C) T ∝ √a D) T ∝ a^(3/2)

34. 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?

A) MR²/2 B) MR² C) 2/5 MR² D) 1/2 MR²

35. What is the moment of inertia of a thin hoop of mass M and radius R about a diameter?

A) MR²/2 B) MR² C) 2/5 MR² D) 1/2 MR²

36. 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:

A) mvr B) mv/r C) mr/v D) mvr²

37. If the central force is F(r) = -k r, what is the shape of the orbit?

A) Parabola B) Hyperbola C) Ellipse or Circle D) Straight line

38. The moment of inertia of a body depends on which of the following factors?

A) Mass distribution and axis of rotation B) Total mass and velocity C) Shape and linear momentum D) Volume and temperature

39. Which of the following shapes has the smallest moment of inertia for the same mass and radius, about an axis through its center?

A) Solid sphere B) Solid cylinder C) Thin spherical shell D) Thin hoop

40. A particle is moving in a central force field. If its total energy is negative, the orbit is:

A) Elliptical B) Parabolic C) Hyperbolic D) Rectilinear

41. If the total energy of a particle in a central force field is zero, the orbit is:

A) Elliptical B) Parabolic C) Hyperbolic D) Circular

42. If the total energy of a particle in a central force field is positive, the orbit is:

A) Elliptical B) Parabolic C) Hyperbolic D) Closed

43. The radial velocity of a particle in a central orbit is zero when the particle is at:

A) The point of closest approach (periapsis). B) The point of farthest approach (apoapsis). C) Both periapsis and apoapsis. D) Any point on the orbit.

44. 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?

A) (m1+m2)r²/4 B) (m1*m2)r²/ (m1+m2) C) (m1*m2)r²/4 D) (m1+m2)r²/2

45. 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?

A) The force at distance r B) The force as a function of u C) The potential energy D) The kinetic energy

46. 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?

A) dV_eff/dr = 0 and d²V_eff/dr² > 0 B) dV_eff/dr = 0 and d²V_eff/dr² < 0 C) dV_eff/dr > 0 D) dV_eff/dr < 0

47. 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?

A) mv² B) mvr C) mr²/v D) mv²/r

48. 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?

A) Contact force B) Conservative force C) Central force D) Frictional force

49. 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:

A) Ellipse B) Circle C) Parabola D) Hyperbola

Central Orbits and Moment of Inertia - Online Test

Test Results