Question

A particle is moving in a circular path with a constant speed \(v\). If \(\theta\) is the angular displacement. Then starting from \(\theta = 0\), the maximum and minimum changes in the momentum will occur, when value of \(\theta\) is respectively:

• A. \(45^\circ \text{ and } 90^\circ\)
• B. \(90^\circ \text{ and } 180^\circ\)
• C. \(180^\circ \text{ and } 360^\circ\)
• D. \(90^\circ \text{ and } 270^\circ\)

Hint:

Minimum change in momentum occurs when initial and final momenta are same → full revolution.

Answer 1

The Correct Option is C

Concept: \[ \Delta p = 2mv \sin\left(\frac{\theta}{2}\right) \]

Step 1: Maximum change in momentum.
\[ \Delta p \text{ is maximum when } \sin\left(\frac{\theta}{2}\right)=1 \] \[ \Rightarrow \frac{\theta}{2} = 90^\circ \Rightarrow \theta = 180^\circ \]

Step 2: Minimum change in momentum.
\[ \Delta p \text{ is minimum when } \sin\left(\frac{\theta}{2}\right)=0 \] \[ \Rightarrow \frac{\theta}{2} = 180^\circ \Rightarrow \theta = 360^\circ \] Final Answer: \[ 180^\circ \text{ and } 360^\circ \]