Q.Underline the correct alternative:

Concept: Conservative Force Work — For a conservative force, work done equals the negative change in potential energy: $W = -\Delta U$.

1. Positive work ($W > 0$) means $\Delta U < 0$, so potential energy decreases.
2. Friction is non-conservative and dissipative — work done against friction converts kinetic energy into heat, so the body loses kinetic energy.
3. Newton’s second law for a system: $\frac{d\vec{P}}{dt} = \vec{F}_{\text{ext}}$. Internal forces cancel pairwise, so the rate depends only on the external force. …

The Core Idea: Conservative Forces and Potential Energy

A conservative force is one where the work done in moving a particle between two points is independent of the path taken. Gravity and the spring force are classic examples. The key relationship is:

This means: if the conservative force does positive work on the body, then $\Delta U$ must be negative — the potential energy decreases. Think of a ball falling under gravity: gravity does positive work, and the ball’s gravitational potential energy drops. Conversely, if you lift the ball against gravity, you do positive work on the ball, but gravity does negative work, and potential energy increases.

Step-by-Step Analysis

1. Part (a): Conservative force does positive work

When a conservative force does positive work, energy is being transferred from potential energy to kinetic energy. The potential energy reservoir is being drained.

• Positive work by conservative force $\implies W > 0$
• From $W = -\Delta U$, we get $-\Delta U > 0 \implies \Delta U < 0$
• So potential energy decreases.

Answer for (a): decreases

2. Part (b): Work done against friction

Friction is a non-conservative force. When a body does work against friction, it means the body is moving and friction opposes the motion. The body must expend energy to overcome this opposition.

• The work done against friction converts kinetic energy into heat (thermal energy).
• Potential energy is stored energy due to position or configuration — friction does not store energy; it dissipates it.
• Therefore, the loss is of kinetic energy, not potential energy.

Answer for (b): kinetic energy

3. Part (c): Rate of change of total momentum of a many-particle system

Newton’s second law for a system of particles states:

where $\mathbf{P}_{\text{total}}$ is the total linear momentum of the system. Why? Internal forces between particles come in action-reaction pairs. By Newton’s third law, these pairs cancel out when summed over the entire system. So internal forces contribute zero net force on the system.

• The rate of change of total momentum depends only on the net external force.
• The sum of internal forces is always zero for the system as a whole. …

Concept: Core Relations — Conservative Forces, Friction, System Momentum, Collisions

Step 1 (a): Apply $W=-\Delta U$ for a conservative force.

Positive work ($W>0$) means $\Delta U<0$, so potential energy decreases.

Step 2 (b): Identify what friction dissipates.

Friction is non-conservative; work done against it converts kinetic energy (not potential energy) into heat.

Step 3 (c): Apply Newton's second law to a system of particles.

$\frac{d\vec P}{dt}=\vec F_{\text{ext}}$ — internal forces cancel in pairs (Newton's third law), so only the external force matters. …