Motion of Centre of Mass - UNSOLVED PRACTICE SET
Chapter: System of Particles and Rotational Motion | Topic: Motion of Centre of Mass
MOTION OF CENTRE OF MASS - UNSOLVED PRACTICE SET
Topic: Motion of Centre of Mass
Multiple Choice Questions
Q1. The velocity of the centre of mass of a system is given by:
- V_cm = Σmᵢvᵢ
- V_cm = (Σmᵢvᵢ)/M
- V_cm = (Σvᵢ)/n
- V_cm = Σpᵢ
Q2. If no external force acts on a system, the velocity of the centre of mass:
- Increases
- Decreases
- Remains constant
- Becomes zero
Q3. The acceleration of the centre of mass of a system is given by:
- A_cm = F_internal/M
- A_cm = F_external/M
- A_cm = F_total/M
- A_cm = 0
Q4. Two skaters initially at rest push each other apart on frictionless ice. Their centre of mass:
- Moves in the direction of the heavier skater
- Moves in the direction of the lighter skater
- Remains at rest
- Accelerates
Q5. An exploding bomb breaks into three fragments. After explosion, the centre of mass of the fragments:
- Falls vertically downward
- Moves horizontally
- Remains at the explosion point
- Follows the path the bomb would have followed without explosion
Q6. When a boy sitting on a stationary cart in your school playground jumps off horizontally, the cart:
- Remains stationary
- Moves in the same direction as the boy
- Moves in the opposite direction to the boy
- Topples over
Short Answer Questions
Q7. State the condition under which the velocity of the centre of mass of a system remains constant. What happens to the centre of mass if the net external force is zero?
Q8. Two particles of masses 2 kg and 3 kg move with velocities 4 m/s and 2 m/s respectively in the same direction. Calculate the velocity of their centre of mass.
Q9. Explain why the centre of mass of a projectile follows a parabolic path even if it explodes in mid-air.
Q10. A man standing on a frictionless platform throws a ball horizontally. What happens to the man? Explain using the concept of centre of mass.
Q11. Two blocks of masses m₁ and m₂ connected by a spring rest on a frictionless surface. The spring is compressed and released. Describe the motion of the centre of mass.
Q12. A rocket is moving in gravity-free space. When it ejects exhaust gases backward, what happens to the velocity of the centre of mass of the rocket-gas system? Explain.
Long Answer Questions
Q13. Derive the expressions for:
(i) Velocity of centre of mass: V_cm = (Σmᵢvᵢ)/M
(ii) Acceleration of centre of mass: A_cm = F_external/M
Show that the internal forces do not affect the motion of the centre of mass. Explain the physical significance of this result with examples.
Q14. A projectile of mass M is fired at angle θ with velocity v₀. At the highest point, it explodes into two fragments of masses M/3 and 2M/3. The lighter fragment comes to rest immediately after explosion.
(i) Where does the centre of mass of the system strike the ground?
(ii) Where does the heavier fragment strike the ground?
(iii) Calculate the velocity of the heavier fragment just after explosion.
(iv) Explain why the explosion does not affect the motion of the centre of mass.
Q15. Two blocks A (mass 2 kg) and B (mass 3 kg) are placed on a frictionless horizontal surface. Block A is given an initial velocity of 6 m/s towards B. They collide and stick together.
(i) Calculate the velocity of the centre of mass before collision.
(ii) Calculate the velocity of the centre of mass after collision.
(iii) Calculate the final velocity of the combined mass.
(iv) Calculate the loss in kinetic energy. Where does this energy go?
(v) If the blocks had collided elastically instead, how would the velocity of the centre of mass change?
Numerical / Application-Based Problems
Q16. Three particles of masses 1 kg, 2 kg, and 3 kg have position vectors (i + j), (2i – j), and (i + 2j) metres respectively at t = 0. Their velocities are (2i), (i + j), and (i – j) m/s respectively.
(i) Find the position of the centre of mass at t = 0.
(ii) Find the velocity of the centre of mass.
(iii) Find the position of the centre of mass at t = 2 s.
(iv) If an external force F = (3i + 2j) N acts on the system, find the acceleration of the centre of mass.
(v) Find the position of the centre of mass at t = 2 s under this external force, assuming it starts from rest.
Q17. A shell of mass 5 kg is fired from a cannon with velocity 200 m/s at 60° to the horizontal. At the highest point, it explodes into two fragments of masses 2 kg and 3 kg. The 2 kg fragment moves vertically upward with velocity 50 m/s.
(i) Calculate the velocity of the centre of mass just before explosion.
(ii) Calculate the velocity of the 3 kg fragment just after explosion.
(iii) Calculate the maximum height reached by the centre of mass.
(iv) Calculate the horizontal distance traveled by the centre of mass before hitting the ground.
(v) Calculate where each fragment lands relative to the cannon.
Q18. In a school physics demonstration, a student sits on a cart of mass 20 kg that can move on a frictionless horizontal track. The student has mass 50 kg. Initially, both are at rest.
(i) The student walks on the cart with velocity 2 m/s relative to the cart. Calculate the velocity of the cart relative to the ground.
(ii) Calculate the velocity of the centre of mass of the system.
(iii) The student now jumps off the cart horizontally with velocity 3 m/s relative to the cart. Calculate the velocity of the cart after the jump.
(iv) If the student jumps at an angle of 30° to the horizontal, how does the cart's velocity change?
(v) Explain why the centre of mass of the student-cart system remains at rest throughout if no external horizontal force acts.