A particle of mass m moves in a circular orbit under the central potential field, U(r) = - C/r, where C is a positive constant. The correct radius - velocity graph of the particle's motion is :

Solution

Physics sir · 4 steps

Turn the potential into a force, balance it with centripetal force, and see how r depends on v.

1. Force from potential. The force is F = -dU/dr = -C/r². The minus sign means it points towards the centre, so it is attractive with magnitude C/r², like gravity.
2. Circular orbit. This attraction supplies the centripetal force, so mv²/r = C/r². Cancel one r and solve for r.

   r = C/(m v²)

3. Read the graph. r ∝ 1/v². As v grows, r falls and approaches the v-axis without touching it. As v goes towards zero, r shoots up without limit.
4. So. The graph is a decreasing curve that hugs both axes without meeting them, which is option (b).

The idea to keep. For any 1/r potential, faster orbits are tighter orbits: v² ∝ 1/r. Always get the force from -dU/dr, then balance it with mv²/r.

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