A heavy particle of weight w, attached to a fixed point by a light inextensible string describes a circle in a vertical plane. The tension in the string has the values nw and mw respectively when the particle is at the highest and lowest points in the path. Then :

Options

A. `m + n = 6`
B. `m/n = 2`
C. `m - n = -6`
D. `n - m = -6`

Explanation

To solve the problem, we need to analyze the forces acting on the particle at the highest and lowest points of its circular motion and apply the principles of energy conservation.

### Step-by-Step Solution:

1. **Identify Forces at the Lowest Point:**
At the lowest point of the circular path, the forces acting on the particle are:
- The weight \( W \) acting downwards.
- The tension \( T_A \) in the string acting upwards.

The net centripetal force required for circular motion is provided by the difference between the tension and the weight:
\[ T_A - W = \frac{m u^2}{r} \]
where \( u \) is the velocity at the lowest point, \( m \) is the mass of the particle, and \( r \) is the radius of the circular path.

Given that \( T_A = Mw \) (where \( M \) is a constant), we can rewrite the equation:
\[ Mw - W = \frac{W}{g} \cdot \frac{u^2}{r} \]
Simplifying gives us:
\[ (M - 1)W = \frac{W}{g} \cdot \frac{u^2}{r} \]
Dividing both sides by \( W \):
\[ M - 1 = \frac{u^2}{g r} \]
(Equation 1)

2. **Identify Forces at the Highest Point:**
At the highest point of the circular path, the forces acting on the particle are:
- The weight \( W \) acting downwards.
- The tension \( T_B \) in the string also acting downwards.

The net centripetal force required for circular motion is provided by the sum of the weight and the tension:
\[ T_B + W = \frac{m v^2}{r} \]
Given that \( T_B = Nw \) (where \( N \) is another constant), we can rewrite the equation:
\[ Nw + W = \frac{W}{g} \cdot \frac{v^2}{r} \]
Simplifying gives us:
\[ (N + 1)W = \frac{W}{g} \cdot \frac{v^2}{r} \]
Dividing both sides by \( W \):
\[ N + 1 = \frac{v^2}{g r} \]
(Equation 2)

3. **Apply Conservation of Energy:**
By the conservation of mechanical energy, the total mechanical energy at the lowest point must equal the total mechanical energy at the highest point:
\[ \frac{1}{2} m u^2 + 0 = \frac{1}{2} m v^2 + 2mg \]
Simplifying gives:
\[ \frac{1}{2} u^2 = \frac{1}{2} v^2 + 2g \]
Rearranging gives:
\[ u^2 - v^2 = 4g \]
(Equation 3)

4. **Combine Equations:**
From Equation 1 and Equation 2, we have:
\[ M - 1 = \frac{u^2}{g r} \quad \text{and} \quad N + 1 = \frac{v^2}{g r} \]
We can express \( u^2 \) and \( v^2 \) in terms of \( g \) and \( r \):
\[ u^2 = (M - 1) g r \quad \text{and} \quad v^2 = (N + 1) g r \]

5. **Substitute into Energy Equation:**
Substitute \( u^2 \) and \( v^2 \) into Equation 3:
\[ (M - 1) g r - (N + 1) g r = 4g \]
Dividing through by \( g r \) gives:
\[ M - 1 - N - 1 = \frac{4}{r} \]
Rearranging gives:
\[ M - N - 2 = \frac{4}{r} \]

6. **Final Result:**
Rearranging gives:
\[ M - N = 2 + \frac{4}{r} \]