Write the differences between conservative and Non-conservative forces. Give two examples each.

II. Short Answer Questions · Q3

Q. Write the differences between conservative and Non-conservative forces. Give two examples each.

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Concept understanding — Conservative Force Work

What is a Conservative Force? — Starting from Intuition

Imagine you are carrying a bucket of water up a hill and then walking back down. The work your muscles do against gravity depends only on how high you climbed, not on the path you took. Whether you go straight up the steep side or take a long winding road, the net work done by gravity on the bucket when you return to the starting point is exactly zero.

That is the core idea: a force is conservative if the work it does on an object moving between two points depends only on those points, not on the path taken.

The Precise Statement

\vec{F}

Mathematically:

W_{A \to B} = \int_{A}^{B} \vec{F} \cdot d\vec{r} \quad \text{is path-independent}

This single property leads to two equivalent, powerful consequences:

Work around any closed loop is zero. If you go from $A$ to $B$ along one path and return along another, the total work is zero:

\oint \vec{F} \cdot d\vec{r} = 0

The force can be written as the negative gradient of a potential energy function $U$ :

\vec{F} = -\nabla U

This means you can define a potential energy for the system — a stored energy that depends only on position.

Examples You Already Know

Force	Conservative?	Why
Gravity (near Earth)	Yes	Work depends only on height difference
Spring force ( $F = -kx$ )	Yes	Work depends only on stretch/compression
Electrostatic force	Yes	Work depends only on charge positions
Friction	No	Work depends on path length — longer path = more work
Air resistance	No	Same reason — dissipative

Watch out

A common mistake: thinking "conservative" means the force conserves kinetic energy. It does not. It means the force itself allows a potential energy to be defined, so total mechanical energy (kinetic + potential) is conserved when only conservative forces act.

Why This Matters for Exams

When you see a problem with gravity, springs, or electric fields, you can immediately:

Use energy conservation: $K_i + U_i = K_f + U_f$
Ignore the path — only initial and final positions matter
Compute work as $W = -\Delta U$ instead of doing a line integral …

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Step 1. A force is conservative if the work done in moving a body between two points depends only on the two endpoints, not the path taken; it is non-conservative if the work done does depend on the path.

Step 2. Key differences (Table 4.3): work by a conservative force is path-independent, while a non-conservative force's work depends on the path; work around a closed loop is zero for a conservative force but generally nonzero for a non-conservative one; total mechanical energy is conserved under conservative forces but dissipated (as heat) under non-conservative ones; a conservative force can be written as $F=-dU/dx$ , while no such potential-energy relation exists for a non-conservative force.

Step 3. Examples of conservative forces: gravitational force, and the elastic spring (restoring) force. …

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Q.Write the differences between conservative and Non-conservative forces. Give two examples each.

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