1. Can the centre of mass of a body be at a point outside the body?
Answer: Yes
Explanation: The centre of mass (COM) is the point where the entire mass of a body can be considered to be concentrated for the purpose of analyzing motion. It depends on the distribution of mass and not necessarily on the physical boundaries of the body.
- For uniform, symmetric bodies (like a solid sphere or cube), the COM lies inside the body.
- For hollow or ring-shaped bodies, the COM lies at the geometric centre, which is outside the material of the body.
- Example: In a ring or hollow sphere, the COM is at the centre of the ring/sphere, a point where there is no actual matter.
2. If all the particles of a system lie in X–Y plane, is it necessary that the centre of mass be in X–Y plane?
Answer: Yes
Explanation:
The centre of mass (COM) is defined as the weighted average of the positions of all particles in the system. Mathematically,
\[
\vec{R}_{CM} = \frac{\sum m_i \vec{r}_i}{\sum m_i}
\]
If every particle has coordinates \((x_i, y_i, 0)\), i.e. all lie in the X–Y plane, then:
\[
\vec{R}_{CM} = \left(\frac{\sum m_i x_i}{M}, \frac{\sum m_i y_i}{M}, \frac{\sum m_i \cdot 0}{M}\right)
\]
\[
\vec{R}_{CM} = \left(X_{CM}, Y_{CM}, 0\right)
\]
Thus, the z-coordinate of the COM is zero, meaning the COM also lies in the X–Y plane.
So, whenever all particles of a system are confined to a plane, the centre of mass must lie in the same plane.
3. If all the particles of a system lie in a cube, is it necessary that the centre of mass be in the cube?
Answer: No, not necessarily.
Explanation: The centre of mass (COM) is the weighted average of the positions of all particles. If all particles are confined inside a cube, the COM will lie within the convex hull of those particles.
- If the cube is solid and uniformly filled, the COM will indeed lie inside the cube.
- However, if the particles are arranged only at certain corners or edges, the COM can shift outside the cube.
Example:
- Suppose we place three equal masses at three corners of a cube. The COM will lie at the centroid of those three points, which may fall outside the cube.
- Similarly, if particles are distributed along one face or edge, the COM can lie outside the cube’s volume, though still within the geometric region defined by the particles.
So, unlike the X–Y plane case (where confinement guarantees the COM lies in the plane), confinement inside a cube does not guarantee that the COM lies inside the cube itself.
4. The centre of mass is defined as \( R = \frac{1}{M} \sum m_i r_i \). Suppose we define “centre of charge” as \( R_C = \frac{1}{Q} \sum q_i r_i \) where \( q_i \) represents the ith charge placed at \( r_i \) and \( Q \) is the total charge of the system. (a) Can the centre of charge of a two-charge system be outside the line segment joining the charges? (b) If all the charges of a system are in X-Y plane, is it necessary that the centre of charge be in X-Y plane? (c) If all the charges of a system lie in a cube, is it necessary that the centre of charge be in the cube?
Answer:
- (a) Yes. Unlike mass, charge can be negative. If the charges have opposite signs, the centre of charge can lie outside the line segment joining them.
- (b) Yes. If all charges have \( z = 0 \), then the calculated centre of charge will also have \( z = 0 \), hence it lies in the X-Y plane.
- (c) No. Because charges can be negative, the weighted average is not restricted to the interior of the cube.
5. The weight \(Mg\) of an extended body is generally shown in a diagram to act through the centre of mass. Does it mean that the earth does not attract other particles?
Answer: No, it does not mean that.
Explanation:
- The earth attracts every particle of the body individually with a gravitational force.
- However, when analyzing the motion of the whole body, it is convenient to represent the
net gravitational force as a single force acting at the
centre of mass (COM). - This is a simplification: instead of drawing many small forces on each particle, we replace them with
one equivalent force \(Mg\) acting at the COM. - This works because the gravitational field near the Earth’s surface is
uniform (same magnitude and direction for all particles). In such a uniform field,
the resultant force is exactly equal to the total weight acting at the COM.
So, the diagram showing weight at the COM is a representation tool, not a literal statement that Earth ignores other particles.
Earth’s gravity acts on every particle, but the combined effect can be treated as if it acts at the COM.
6. A bob suspended from the ceiling of a car which is accelerating on a horizontal road. The bob stays at rest with respect to the car with the string making an angle θ with the vertical. The linear momentum of the bob as seen from the road is increasing with time. Is it a violation of conservation of linear momentum? If not, where is the external force which changes the linear momentum?
Answer: No, it is not a violation of conservation of linear momentum.
Explanation: The law of conservation of linear momentum holds only for an isolated system, i.e., one on which no external force acts. In this case, the bob is not isolated — it is connected to the car by a string.
As the car accelerates forward, the string exerts a horizontal tension on the bob. This tension provides the external force that changes the bob’s momentum with respect to the road.
From the car’s frame, the bob appears stationary because the tension balances the pseudo‑force due to acceleration. But from the road’s frame, the bob’s velocity and hence its linear momentum increase with time, caused by the horizontal component of tension acting on it.
Therefore, the increase in momentum is due to an external force (tension in the string), and there is no violation of the conservation of linear momentum.
7. You are waiting for a train on a railway platform. Your three‑year‑old niece is standing on your iron trunk containing the luggage. Why does the trunk not recoil as she jumps off on the platform?
Answer: The trunk does not recoil because the external force of friction between the trunk and the ground prevents it from moving.
Explanation: When your niece jumps off, she exerts a downward and backward force on the trunk. According to Newton’s third law, the trunk exerts an equal and opposite force on her. If the trunk were on a smooth, frictionless surface, this backward force would make it recoil slightly. However, on the rough platform, the static friction between the trunk and the ground provides an external force that opposes this motion. This frictional force prevents the trunk from moving, so its momentum remains unchanged.
Hence, there is no violation of conservation of momentum, because the system (trunk + Earth) is not isolated — the Earth exerts friction on the trunk, absorbing the tiny impulse.
8. In a head‑on collision between two particles, is it necessary that the particles will acquire a common velocity at least for one instant?
Answer: No, it is not necessary.
Explanation: A head‑on collision means the particles move along the same straight line before and after impact. Whether they acquire a common velocity at any instant depends on the nature of the collision:
- In a perfectly elastic collision, the particles exchange momentum and energy but do not necessarily have the same velocity at any instant. They approach each other, interact, and then separate with different velocities.
- In a perfectly inelastic collision, the particles stick together after impact and move with a common velocity immediately after collision.
- In partially elastic collisions, they may momentarily have the same velocity during contact, but this is not guaranteed—it depends on the details of the interaction forces and masses.
Thus, acquiring a common velocity is not a necessary condition for all head‑on collisions; it occurs only in completely inelastic collisions.
9. A collision experiment is done on a horizontal table kept in an elevator. Do you expect a change in the results if the elevator is accelerated up or down because of the noninertial character of the frame?
Answer: No, the results of the collision experiment will not change.
Explanation: When the elevator accelerates upward or downward, the effective acceleration due to gravity inside the elevator changes:
- If the elevator accelerates upward, effective gravity becomes .
- If the elevator accelerates downward, effective gravity becomes .
However, in a horizontal collision experiment, the motion and momentum exchange occur along the horizontal direction, while gravity (and its variation) acts vertically. Thus, the vertical change in effective gravity does not affect the horizontal motion or the results of the collision — provided the table is smooth and horizontal.
The only effect is a change in the normal reaction between the table and the bodies, but this does not influence the horizontal momentum conservation or the collision outcomes.
Hence, the results of the collision experiment remain unchanged, even though the elevator is a noninertial frame.
10. Two bodies make an elastic head‑on collision on a smooth horizontal table kept in a car. Do you expect a change in the result if the car is accelerated on a horizontal road because of the noninertial character of the frame? Does the equation “Velocity of separation = Velocity of approach” remain valid in an accelerating car? Does the equation “final momentum = initial momentum” remain valid in the accelerating car?
Answer: Yes, the results will change if the car is accelerated, because the car becomes a noninertial frame.
Explanation:
- When the car accelerates, a pseudo force acts on each body opposite to the direction of acceleration.
- This pseudo force is external to the two‑body system (from the car’s frame viewpoint).
- Therefore, the system is not isolated, and the law of conservation of momentum does not hold in the accelerating frame.
- The equation “final momentum = initial momentum” is valid only in an inertial frame, so it fails in the accelerating car.