A particle of mass is made to move with uniform speed v along the perimeter of a regular polygon of 2n sides. The magnitude of impulse of impulse applied at each corner of the polygon in amv sin `pi//bn`. Then a/b =

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to analyze the motion of a particle moving along the perimeter of a regular polygon with 2n sides and determine the impulse applied at each corner.

### Step-by-Step Solution:

1. **Understanding the Polygon and Motion**:
- A regular polygon with \(2n\) sides means it has \(2n\) corners.
- The particle moves with a uniform speed \(v\) along the perimeter of this polygon.

2. **Initial and Final Momentum**:
- The initial momentum \(p_i\) of the particle at a corner is given by:
\[ p_i = mv \]
- After the particle turns at the corner, the final momentum \(p_f\) will also be:
\[ p_f = mv \]

3. **Angle Between Momentum Vectors**:
- The angle \(\theta\) between the initial and final momentum vectors at each corner is \(2\pi/n\) because the internal angle between two sides of a regular polygon with \(2n\) sides is \(2\pi/n\).

4. **Calculating the Impulse**:
- The impulse \(J\) can be calculated using the formula:
\[ J = p_f - p_i \]
- In terms of magnitudes, the impulse can also be expressed as:
\[ |J| = |p_f| + |p_i| - 2|p_f||p_i|\cos(\theta) \]
- Substituting \(p_f = mv\), \(p_i = mv\), and \(\theta = \frac{2\pi}{n}\):
\[ |J| = mv + mv - 2(mv)(mv)\cos\left(\frac{2\pi}{n}\right) \]
- This simplifies to:
\[ |J| = 2mv - 2m^2v^2\cos\left(\frac{2\pi}{n}\right) \]

5. **Using Trigonometric Identity**:
- We can use the identity \(1 - \cos(x) = 2\sin^2(x/2)\):
\[ |J| = 2mv\left(1 - \cos\left(\frac{2\pi}{n}\right)\right) = 2mv\cdot 2\sin^2\left(\frac{\pi}{n}\right) \]
- Therefore, we have:
\[ |J| = 4mv\sin^2\left(\frac{\pi}{n}\right) \]

6. **Comparing with Given Expression**:
- The problem states that the impulse is given by:
\[ |J| = amv\sin\left(\frac{\pi}{bn}\right) \]
- From our derived expression, we can equate:
\[ 4mv\sin^2\left(\frac{\pi}{n}\right) = amv\sin\left(\frac{\pi}{bn}\right) \]

7. **Finding the Ratio \(a/b\)**:
- We can see that \(a = 4\) and \(b = 1\) (since \(\sin^2\) can be rewritten in terms of \(\sin\)).
- Thus, the ratio \(a/b\) is:
\[ \frac{a}{b} = \frac{4}{1} = 4 \]

### Final Answer:
\[ \frac{a}{b} = 4 \]