Which one of the following forces cannot be expressed in terms of potential energy?

Solution

Physics sir · 3 steps

A potential energy exists only if the work done depends on the end points, not on the path taken.

1. When PE exists. A force can be written as F = -dU/dx only if it is conservative. That means the work it does between two points is the same along every path, and it is zero around any closed loop.

   Force	Work depends on path?	Potential energy
   Frictional	Yes, a longer path means more work against friction	None
   Coulomb	No	U = kq₁q₂/r
   Restoring (spring)	No	U = 1/2kx²
   Gravitational	No	U = -GMm/r

2. Why friction fails. Friction always opposes the motion, so it does negative work on every part of the path. Even on a closed loop the total work is not zero, and the lost energy turns into heat. No function U(x) can describe that.
3. So. Friction is non-conservative and cannot be expressed through a potential energy. The answer is (a).

The idea to keep. Ask one question: does the work around a closed loop come out zero? If yes, the force is conservative and has a potential energy. Friction and air drag never pass this test.

Answer from the NTA final answer key. Solution by PrepWiser.