Angular Momentum MCQs
1. A round disc of moment of inertia I₂ about its axis perpendicular to its plane and passing through its centre is placed over another disc of moment of inertia I₁ rotating with an angular velocity ω about the same axis. The final angular velocity of the combination of discs is
Correct Answer: (a) \(\frac{I_1 \omega}{I_1 + I_2}\)
Using conservation of angular momentum: I₁ω = (I₁ + I₂)ω' ⇒ ω' = \(\frac{I_1 \omega}{I_1 + I_2}\)
2. The angular momentum of a system of particles is conserved
Correct Answer: (b) When no external torque acts on the system
According to the principle of conservation of angular momentum, angular momentum is conserved when the net external torque acting on the system is zero.
3. The motion of planets in the solar system is an example of the conservation of
Correct Answer: (c) Angular momentum
Planetary motion follows Kepler's second law, which is a consequence of conservation of angular momentum.
4. A thin and circular disc of mass M and radius R is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with an angular velocity ω. If another disc of same dimensions but of mass M/4 is placed gently on the first disc co-axially, then the new angular velocity of the system is
Correct Answer: (b) \(\frac{4\omega}{5}\)
Moment of inertia of first disc I₁ = ½MR², second disc I₂ = ½(M/4)R² = MR²/8. Using conservation of angular momentum: I₁ω = (I₁ + I₂)ω' ⇒ (½MR²)ω = (½MR² + MR²/8)ω' ⇒ ω' = \(\frac{4\omega}{5}\)
5. A man is sitting on a rotating table with his arms stretched outwards. When he suddenly folds his arms inside, then his
Correct Answer: (c) Moment of inertia decreases
When the man folds his arms, his mass distribution changes, reducing his moment of inertia. Angular momentum is conserved, so angular velocity increases.
6. A ballet dancer, dancing on a smooth floor is spinning about a vertical axis with her arms folded with an angular velocity of 20 rad/s. When she stretches her arms fully, the spinning speed decreases to 10 rad/s. If I is the initial moment of inertia of the dancer, the new moment of inertia is
Correct Answer: (a) 2I
Using conservation of angular momentum: I₁ω₁ = I₂ω₂ ⇒ I × 20 = I₂ × 10 ⇒ I₂ = 2I
7. The rotational kinetic energy of a body is E and its moment of inertia is I. The angular momentum is
Correct Answer: (b) \(\sqrt{2EI}\)
Rotational kinetic energy E = ½Iω² ⇒ ω = \(\sqrt{\frac{2E}{I}}\). Angular momentum L = Iω = I × \(\sqrt{\frac{2E}{I}}\) = \(\sqrt{2EI}\)
8. A disc is rotating with an angular speed of ω. If a child sits on it, which of the following is conserved
Correct Answer: (d) Angular momentum
When the child sits on the rotating disc, no external torque acts on the system, so angular momentum is conserved.
9. Torque applied on a particle is zero, then its angular momentum will be
Correct Answer: (c) Both (a) and (b)
When torque is zero, angular momentum is conserved in both magnitude and direction.
10. Angular momentum of a system of particles changes when
Correct Answer: (b) Torque acts on a body
According to the rotational analog of Newton's second law, torque equals the rate of change of angular momentum: τ = dL/dt.
11. A thin circular ring of mass M and radius r is rotating about its axis with a constant angular velocity ω. Four objects each of mass m, are kept gently to the opposite ends of two perpendicular diameters of the ring. The angular velocity of the ring will be
Correct Answer: (a) \(\frac{M\omega}{M+4m}\)
Initial angular momentum L₁ = Mr²ω. After placing masses, moment of inertia becomes Mr² + 4mr². Using conservation of angular momentum: Mr²ω = (M+4m)r²ω' ⇒ ω' = \(\frac{M\omega}{M+4m}\)
12. If a force acts on a body at a point away from the centre of mass, then
Correct Answer: (c) Both change
A force acting away from the center of mass produces both linear acceleration (a = F/m) and angular acceleration (α = τ/I).
13. The moment of momentum is called
Correct Answer: (d) Angular momentum
Angular momentum is defined as the moment of linear momentum about a point.
14. If a gymnast, sitting on a rotating stool with his arms outstretched, suddenly lowers his hands
Correct Answer: (b) His moment of inertia decreases
When the gymnast lowers his hands, his mass distribution changes, reducing his moment of inertia. Angular momentum is conserved, so angular velocity increases.
15. Two discs of moment of inertia I₁ and I₂ and angular speeds ω₁ and ω₂ are rotating along collinear axes passing through their centre of mass and perpendicular to their plane. If the two are made to rotate combinedly along the same axis the rotational KE of system will be
Correct Answer: (a) \(\frac{(I_1\omega_1 + I_2\omega_2)^2}{2(I_1+I_2)}\)
Using conservation of angular momentum: I₁ω₁ + I₂ω₂ = (I₁+I₂)ω. Final KE = ½(I₁+I₂)ω² = ½(I₁+I₂)\(\left(\frac{I_1\omega_1 + I_2\omega_2}{I_1+I_2}\right)^2\) = \(\frac{(I_1\omega_1 + I_2\omega_2)^2}{2(I_1+I_2)}\)
16. The principle of conservation of angular momentum, states that angular momentum
Correct Answer: (d) None of these
Angular momentum is conserved only when the net external torque acting on the system is zero.
17. The position of a particle is given by: \(\vec{r} = 3\hat{i} + 4\hat{j}\) and momentum \(\vec{p} = 5\hat{i} + 6\hat{j}\). The angular momentum is perpendicular to
Correct Answer: (c) Z-axis
Angular momentum \(\vec{L} = \vec{r} \times \vec{p}\) is perpendicular to both position and momentum vectors, so it's along the Z-axis.
18. A particle of mass m is rotating in a plane in circular path of radius r. Its angular momentum is L. The centripetal force acting on the particle is
Correct Answer: (b) \(\frac{L^2}{mr^3}\)
Angular momentum L = mvr ⇒ v = L/mr. Centripetal force F = mv²/r = m(L/mr)²/r = L²/(m²r² × r) = L²/(mr³)
19. A particle performs uniform circular motion with an angular momentum L. If the frequency of a particle's motion is doubled and its kinetic energy is halved, the angular momentum becomes
Correct Answer: (d) L/4
Kinetic energy E = L²/(2I). If E becomes E/2 and frequency doubles, angular velocity ω doubles. Since L = Iω, and E ∝ L², when E is halved, L becomes L/√2. But ω doubles, so I must decrease by factor of 4. Thus L = Iω becomes (I/4)(2ω) = Iω/2 = L/2? Let's calculate properly: E = L²/(2I) = ½Iω². If ω doubles and E halves, then ½I(2ω)² = 2Iω² = E/2 ⇒ I becomes I/4. Then L = Iω becomes (I/4)(2ω) = Iω/2 = L/2. Wait, there's inconsistency. Let's solve systematically: E₁ = L₁²/(2I₁) = ½I₁ω₁². E₂ = E₁/2, ω₂ = 2ω₁. E₂ = ½I₂ω₂² = ½I₂(4ω₁²) = 2I₂ω₁² = E₁/2 = ¼I₁ω₁² ⇒ I₂ = I₁/8. Then L₂ = I₂ω₂ = (I₁/8)(2ω₁) = I₁ω₁/4 = L₁/4.
20. Two rigid bodies A and B rotate with rotational kinetic energies E_A and E_B respectively. The moments of inertia of A and B about the axis of rotation are I_A and I_B respectively. If I_A = I_B/4 and E_A = 100 E_B the ratio of angular momentum (L_A) of A to the angular momentum (L_B) of B is
Correct Answer: (c) 5
E = L²/(2I) ⇒ L = √(2EI). L_A/L_B = √[(2E_AI_A)/(2E_BI_B)] = √[(E_A/E_B)×(I_A/I_B)] = √[100×(1/4)] = √25 = 5
21. What remains constant when the earth revolves around the sun
Correct Answer: (a) Angular momentum
The gravitational force acting on Earth is a central force, so torque is zero and angular momentum is conserved.
22. Before jumping in water from above a swimmer bends his body to
Correct Answer: (b) Decrease moment of inertia
By bending his body, the swimmer reduces his moment of inertia, which increases his angular velocity for the same angular momentum, allowing him to complete rotations more quickly.
23. The angular speed of a body changes from ω₁ to ω₂ without applying a torque but due to change in its moment of inertia. The ratio of radii of gyration in the two cases is
Correct Answer: (c) \(\sqrt{\omega_1} : \sqrt{\omega_2}\)
With no torque, angular momentum is conserved: I₁ω₁ = I₂ω₂ ⇒ mk₁²ω₁ = mk₂²ω₂ ⇒ k₁²/k₂² = ω₂/ω₁ ⇒ k₁/k₂ = \(\sqrt{\omega_2}/\sqrt{\omega_1}\) = \(\sqrt{\omega_1} : \sqrt{\omega_2}\)
24. A circular turn table has a block of ice placed at its centre. The system rotates with an angular speed ω about an axis passing through the centre of the table. If the ice melts on its own without any evaporation, the speed of rotation of the system
Correct Answer: (b) Remains constant at the same value ω
When ice melts at the center, there's no change in mass distribution about the axis of rotation, so moment of inertia remains the same. With no external torque, angular velocity remains constant.
25. When a man stands on a turn-table stretching with two equal loads in hand and rotates. Then he folds his arm. Which of the following statement is correct
Correct Answer: (d) Angular velocity increases
When the man folds his arms, his moment of inertia decreases. Angular momentum is conserved, so angular velocity increases.
26. When a mass is rotating in a plane about a fixed point, its angular momentum is directed along
Correct Answer: (d) The axis of rotation
Angular momentum \(\vec{L} = \vec{r} \times \vec{p}\) is perpendicular to the plane of rotation, i.e., along the axis of rotation.
27. A dancer on ice spins faster when she folds her arms. This is due to
Correct Answer: (c) Constant angular momentum and increase in kinetic energy
When the dancer folds her arms, her moment of inertia decreases. Angular momentum is conserved, so angular velocity increases. Kinetic energy = L²/(2I) increases as I decreases.
28. What remains constant in the field of central force
Correct Answer: (c) Angular momentum
In a central force field, torque is zero, so angular momentum is conserved.
29. Before jumping in water from above a swimmer bends his body to
Correct Answer: (b) Decrease moment of inertia
Same as question 22. By bending his body, the swimmer reduces his moment of inertia, which increases his angular velocity for the same angular momentum.
30. The radius of a rotating disc is suddenly reduced to half without any change in its mass. Then its angular velocity will be
Correct Answer: (a) Four times
Moment of inertia of disc I ∝ R². If R becomes R/2, I becomes I/4. With angular momentum conserved, L = Iω = constant, so ω becomes 4 times.
31. Two bodies have their moments of inertia I and 2I respectively about their axis of rotation. If their kinetic energies of rotation are equal, their angular momentum will be in the ratio
Correct Answer: (d) 1 : \(\sqrt{2}\)
E = L²/(2I) ⇒ L = √(2EI). For equal E, L₁/L₂ = √(I₁/I₂) = √(I/2I) = 1/√2 = 1 : √2
32. A man turns on a rotating table with an angular speed ω. He is holding two equal masses at arm's length. Without moving his arms, he just drops the two masses. How will his angular speed change?
Correct Answer: (c) It will remain equal to ω
When the masses are dropped without moving arms, there's no change in mass distribution about the axis, so moment of inertia remains the same. With no external torque, angular velocity remains constant.
33. A person standing on a rotating platform has his hands lowered. He suddenly out stretch his arms. The angular momentum
Correct Answer: (d) Remains the same
With no external torque, angular momentum is conserved.
34. A ring of mass 10 kg and diameter 0.4m is rotated about its axis. If it makes 2100 revolutions per minute, then its angular momentum will be
Correct Answer: (b) 88 kgm²/s
Moment of inertia of ring I = MR² = 10×(0.2)² = 0.4 kgm². Angular velocity ω = 2πf = 2π×(2100/60) = 2π×35 = 70π rad/s. Angular momentum L = Iω = 0.4×70π = 28π ≈ 88 kgm²/s.
35. If the angular momentum of any rotating body increases by 200%, then the increase in its kinetic energy
Correct Answer: (b) 800%
E = L²/(2I). If L increases by 200%, it becomes 3L. Then E becomes (3L)²/(2I) = 9L²/(2I) = 9E, which is an increase of 800%.
36. A swimmer while jumping into water from a height easily forms a loop in the air, if
Correct Answer: (a) He pulls his arms and legs in
By pulling his arms and legs in, the swimmer decreases his moment of inertia, which increases his angular velocity for the same angular momentum, making it easier to complete loops.
37. A solid sphere is rotating in free space. If the radius of the sphere is increased keeping mass same which one of the following will not be affected
Correct Answer: (b) Angular momentum
In free space with no external torque, angular momentum is conserved regardless of changes in radius.
38. If torque is zero then
Correct Answer: (a) Angular momentum is conserved
When net external torque is zero, angular momentum is conserved.
39. The angular momentum of a system of particles is not conserved
Correct Answer: (b) When a net external torque is acting upon the system
Angular momentum is conserved only when net external torque is zero.
40. In an orbital motion, the angular momentum vector is
Correct Answer: (d) Perpendicular to the orbital plane
Angular momentum \(\vec{L} = \vec{r} \times \vec{p}\) is perpendicular to the plane containing position and momentum vectors, i.e., perpendicular to the orbital plane.
41. A boy comes running and sits on a rotating platform. What is conserved
Correct Answer: (c) Angular momentum
When the boy sits on the rotating platform, no external torque acts on the system, so angular momentum is conserved.
42. Calculate the angular momentum of a body whose rotational energy is 10 joule. If the angular momentum vector coincides with the axis of rotation and its moment of inertia about this axis is 8 × 10⁻⁷ kg m²
Correct Answer: (a) 4 × 10⁻³ kgm²/s
E = L²/(2I) ⇒ L = √(2EI) = √(2×10×8×10⁻⁷) = √(160×10⁻⁷) = √(16×10⁻⁶) = 4×10⁻³ kgm²/s
43. Angular momentum L of body with mass moment of inertia I and angular velocity ω rad/sec is equal to
Correct Answer: (b) Iω
This is the definition of angular momentum: L = Iω
44. If the earth is treated as a sphere of radius R and mass M. Its angular momentum about the axis of rotation with period T is
Correct Answer: (c) \(\frac{4\pi MR^2}{5T}\)
Moment of inertia of sphere I = ⅖MR². Angular velocity ω = 2π/T. Angular momentum L = Iω = ⅖MR² × 2π/T = 4πMR²/(5T)
45. A man is standing at the edge of a circular plate which is rotating with a constant angular speed about a perpendicular axis passing through the centre. If the man walks towards the axis along the radius, its angular velocity.
Correct Answer: (c) Increases
When the man walks towards the center, his distance from the axis decreases, reducing the moment of inertia. With angular momentum conserved, angular velocity increases.
46. If linear velocity is constant then angular velocity is proportional to
Correct Answer: (c) 1/r
v = rω ⇒ ω = v/r. If v is constant, ω ∝ 1/r.
47. A diver in a swimming pool bends his head before diving, because it,
Correct Answer: (a) Decreases his moment of inertia
By bending his head, the diver reduces his moment of inertia, which increases his angular velocity for the same angular momentum.
48. The angular momentum of a particle
Correct Answer: (a) Is perpendicular to the plane of rotation
Angular momentum \(\vec{L} = \vec{r} \times \vec{p}\) is perpendicular to the plane containing position and momentum vectors.
49. If the earth is a point mass of 6 × 10²⁴ kg revolving around the sun at a distance of 1.5 × 10⁸ km and in time T = 3.14 × 10⁷ s, then the angular momentum of the earth around the sun is
Correct Answer: (b) 2.7 × 10⁴⁰ kgm²/s
Treating Earth as point mass, I = MR² = 6×10²⁴×(1.5×10¹¹)² = 1.35×10⁴⁷ kgm². ω = 2π/T = 2π/(3.14×10⁷) = 2×10⁻⁷ rad/s. L = Iω = 1.35×10⁴⁷×2×10⁻⁷ = 2.7×10⁴⁰ kgm²/s.
50. A thin circular ring of mass M and radius R is rotating about its axis with a constant angular velocity ω. Two objects each of mass m, are attached gently to the opposite ends of a diameter of the ring. The ring now rotates with an angular velocity
Correct Answer: (a) \(\frac{M\omega}{M+2m}\)
Initial angular momentum L = MR²ω. After adding masses, moment of inertia becomes MR² + 2mR². Using conservation of angular momentum: MR²ω = (M+2m)R²ω' ⇒ ω' = \(\frac{M\omega}{M+2m}\)