Questions & explanations
1. Which of the following statements correctly distinguishes between the centre of mass and the centre of gravity of a body?
- The centre of mass is the point where the entire mass of the body is concentrated, while the centre of gravity is the point where the entire weight acts.
- The centre of mass and centre of gravity always coincide for any object.
- The centre of gravity depends on the gravitational field strength, whereas the centre of mass depends only on the distribution of mass.
- The centre of mass is a geometric point, while the centre of gravity is a physical point.
Answer: The centre of gravity depends on the gravitational field strength, whereas the centre of mass depends only on the distribution of mass.
The centre of mass is solely a function of mass distribution, independent of external forces such as gravity. In contrast, the centre of gravity is the point where the total weight of the body acts, and its location depends on the gravitational field (e.g., its non‑uniformity). In a uniform gravitational field, both points coincide, but they differ when the field is non‑uniform. Options A and D only restate definitions without true distinction; option B is false because they coincide only in uniform gravity. Hence option C is correct.
2. The rotational analogue of mass in linear motion is:
- Moment of inertia
- Angular momentum
- Torque
- Angular displacement
Answer: Moment of inertia
In linear motion, mass is the measure of inertia. In rotational motion, moment of inertia plays the same role — it resists changes in rotational state. Torque is analogous to force, angular momentum to linear momentum, and angular displacement to linear displacement.
3. Angular momentum of a rigid body rotating about a fixed axis is given by:
- L = Iω
- L = mv
- L = r × p
- L = τ t
Answer: L = Iω
For a rigid body rotating about a fixed axis, the total angular momentum is the product of its moment of inertia and angular velocity (L = Iω). Option B is linear momentum, option C is the definition for a single particle, and option D is angular impulse.
4. The torque acting on a rotating body is equal to:
- rate of change of angular momentum
- product of moment of inertia and angular displacement
- angular momentum times time
- rate of change of linear momentum
Answer: rate of change of angular momentum
Newton's second law for rotation states that the net external torque equals the rate of change of angular momentum (τ = dL/dt). Option B would give angular displacement, not torque; option C is dimensionally incorrect; option D is the linear case (force).
5. A person sitting on a rotating stool with arms outstretched suddenly folds his arms. Which of the following remains conserved?
- Linear momentum
- Angular momentum
- Kinetic energy
- Both angular momentum and kinetic energy
Answer: Angular momentum
No external torque acts on the person-stool system, so angular momentum is conserved. Kinetic energy can change because internal muscular work is done; linear momentum may not be conserved overall. Hence only angular momentum remains constant.
6. Two equal and opposite forces of magnitude F act on a rigid body with a perpendicular distance d between their lines of action. The net torque (moment) of the couple is:
- F d
- 2 F d
- 0
- F d / 2
Answer: F d
A couple consists of two equal and opposite forces whose lines of action are parallel but not coincident. The moment of the couple is given by the product of the magnitude of one force and the perpendicular distance between them: τ = F d.
7. A solid sphere rolls without slipping down an inclined plane of inclination θ. The acceleration of its centre of mass is:
- g sinθ
- (5/7) g sinθ
- (7/5) g sinθ
- (2/3) g sinθ
Answer: (5/7) g sinθ
For a solid sphere, moment of inertia I = (2/5)MR². The acceleration of a body rolling down an incline without slipping is a = g sinθ / (1 + I/(MR²)). Here I/(MR²) = 2/5, so a = g sinθ / (1 + 2/5) = g sinθ / (7/5) = (5/7) g sinθ.
8. A solid sphere of mass M and radius R is rolling without slipping on a horizontal surface with a linear speed v. What is the total kinetic energy of the sphere?
- Mv²
- 7/10 Mv²
- 1/2 Mv²
- 5/7 Mv²
Answer: 7/10 Mv²
For a solid sphere, moment of inertia I = (2/5)MR². In rolling without slipping, ω = v/R. Rotational KE = ½ I ω² = ½ × (2/5 MR²) × (v²/R²) = (1/5) Mv². Translational KE = ½ Mv². Total KE = ½ Mv² + 1/5 Mv² = 7/10 Mv².
9. A force of 10 N is applied in the +y direction at a point located at (3 m, 0, 0). What is the direction of the torque about the origin?
- +x
- +y
- +z
- -z
Answer: +z
Using the right-hand rule, point fingers from r (along +x) towards F (along +y), the thumb points in the direction of the torque, which is +z. Mathematically, τ = r × F = (3 i) × (10 j) = 30 k, i.e., +z direction.
10. On which factor(s) does the moment of inertia of a rigid body depend?
- Mass, mass distribution and axis of rotation
- Mass only
- Mass distribution only
- Axis of rotation only
Answer: Mass, mass distribution and axis of rotation
Moment of inertia I = Σ mᵢrᵢ² depends on the mass of the body, how that mass is distributed (the distances rᵢ), and the chosen axis of rotation (which determines the rᵢ values). Changing any of these changes I.
11. A ballet dancer spins with her arms outstretched. When she pulls her arms closer to her body, her angular velocity:
- decreases
- increases
- remains the same
- becomes zero
Answer: increases
Pulling arms in reduces the moment of inertia. Since no external torque acts, angular momentum (L = Iω) remains constant, so a decrease in I must be compensated by an increase in angular velocity ω.
12. Three point masses, each of mass m, are placed at the vertices of an equilateral triangle of side a. The moment of inertia of the system about an axis perpendicular to the plane of the triangle and passing through one vertex is:
- ma²
- 2ma²
- 3ma²
- 0
Answer: 2ma²
The axis passes through one vertex, so that mass contributes zero (r = 0). The other two vertices are at distance a from the axis, each contributing m a². Total I = 0 + m a² + m a² = 2ma².