A particle of mass `m` is made to move with uniform speed `u` along the perimeter of a regular polygon of `n` sides. What is the magnitude of impulse applied by the particle at each corner of the polygon?

Explanation

To solve the problem of finding the magnitude of impulse applied by a particle of mass `m` moving with uniform speed `u` at each corner of a regular polygon with `n` sides, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Motion**:
- The particle moves along the perimeter of a regular polygon with `n` sides. At each corner, the direction of the particle changes.

2. **Determine the Angle of Turn**:
- The angle turned at each corner of the polygon is given by:
\[ \theta = \frac{2\pi}{n} \]
- This angle represents the change in direction of the particle as it moves from one side of the polygon to the next.

3. **Calculate Initial and Final Momentum**:
- When the particle approaches a corner, its momentum can be expressed as:
\[ \vec{p_i} = m \vec{u} \]
- After the turn at the corner, the momentum becomes:
\[ \vec{p_f} = m \vec{u'} \]
- Since the speed remains constant but the direction changes, we can represent the final velocity vector after the turn.

4. **Using the Impulse-Momentum Theorem**:
- The impulse applied at the corner can be calculated using the change in momentum:
\[ \vec{J} = \Delta \vec{p} = \vec{p_f} - \vec{p_i} \]

5. **Finding the Magnitude of Impulse**:
- The change in momentum can be expressed in terms of the angle of turn:
\[ |\vec{J}| = m |\vec{u} - \vec{u'}| \]
- The magnitude of the impulse can also be derived from the sine of the angle:
\[ |\vec{J}| = 2m u \sin\left(\frac{\theta}{2}\right) = 2m u \sin\left(\frac{\pi}{n}\right) \]

6. **Final Expression**:
- Therefore, the magnitude of the impulse applied by the particle at each corner of the polygon is:
\[ J = 2m u \sin\left(\frac{\pi}{n}\right) \]

### Summary:
The magnitude of the impulse applied by the particle at each corner of a regular polygon with `n` sides, moving with uniform speed `u`, is given by:
\[ J = 2m u \sin\left(\frac{\pi}{n}\right) \]