Two bodies of masses m and M are attached to the two ends of a light string passing over a fixed ideal pulley `(M gt gt m)`. When the bodies are in motion, the tension in the string is approximately

Options

A. `(M-m)g`
B. mg
C. 2 mg
D. (m/M)mg

Answer

To find the tension in the string when two bodies of masses \( m \) and \( M \) are attached to either end of a light string passing over a fixed ideal pulley, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Forces Acting on Each Mass:**
- For mass \( M \) (the larger mass), the forces acting on it are:
- Weight: \( Mg \) (downward)
- Tension: \( T \) (upward)
- For mass \( m \) (the smaller mass), the forces acting on it are:
- Weight: \( mg \) (downward)
- Tension: \( T \) (upward)

2. **Apply Newton's Second Law:**
- For mass \( M \) (going down), the equation of motion can be written as:
\[ Mg - T = Ma \quad \text{(1)} \]
- For mass \( m \) (going up), the equation of motion can be written as:
\[ T - mg = ma \quad \text{(2)} \]

3. **Express Acceleration \( a \):**
- From equation (1):
\[ T = Mg - Ma \quad \text{(3)} \]
- From equation (2):
\[ T = ma + mg \quad \text{(4)} \]

4. **Set Equations for Tension Equal:**
- Set equations (3) and (4) equal to each other:
\[ Mg - Ma = ma + mg \]

5. **Rearrange to Solve for Acceleration \( a \):**
- Rearranging gives:
\[ Mg - mg = Ma + ma \]
- Factor out \( a \):
\[ Mg - mg = a(M + m) \]
- Therefore, the acceleration \( a \) is:
\[ a = \frac{(M - m)g}{M + m} \quad \text{(5)} \]

6. **Substitute \( a \) Back to Find Tension \( T \):**
- Substitute \( a \) from equation (5) into either equation (3) or (4). Using equation (3):
\[ T = Mg - M\left(\frac{(M - m)g}{M + m}\right) \]
- Simplifying gives:
\[ T = Mg - \frac{M(M - m)g}{M + m} \]
- Combine terms:
\[ T = \frac{Mg(M + m) - M(M - m)g}{M + m} \]
- This simplifies to:
\[ T = \frac{(Mg^2 + mgM) - (M^2g - mgM)}{M + m} \]
- Further simplification leads to:
\[ T \approx \frac{2mg}{1} \quad \text{(for } M \gg m\text{)} \]

7. **Final Result:**
- Thus, the tension in the string is approximately:
\[ T \approx 2mg \]