A body of mass `M` is attached to the lower end of a metal wire, whose upper end is fixed . The elongation of the wire is `l`.

Options

A. Loss in gravitational potential energy of` M` is `Mgl`
B. The elastic potential energy stored in the wire is ` Mgl`
C. The elastic potential energy stored in the wire is `1/2 Mgl`
D. Heat produced is `1/2 Mgl`

Explanation

To solve the problem, we need to analyze the situation of a mass \( M \) attached to a metal wire and the elongation \( l \) of the wire. We will calculate the loss in gravitational potential energy, the elastic potential energy stored in the wire, and the heat produced.

### Step-by-Step Solution:

1. **Determine the Loss in Gravitational Potential Energy:**
- When the mass \( M \) is attached to the wire, it falls a distance equal to the elongation \( l \).
- The loss in gravitational potential energy \( \Delta U \) can be calculated using the formula:
\[ \Delta U = M g l \]
where \( g \) is the acceleration due to gravity.

2. **Calculate the Elastic Potential Energy Stored in the Wire:**
- The elastic potential energy \( U_e \) stored in the wire when it is stretched can be expressed as:
\[ U_e = \frac{1}{2} F l \]
- Here, \( F \) is the force applied, which is equal to the weight of the mass \( M \):
\[ F = M g \]
- Substituting \( F \) into the elastic potential energy formula gives:
\[ U_e = \frac{1}{2} (M g) l = \frac{1}{2} M g l \]

3. **Determine the Heat Produced:**
- According to the law of conservation of energy, the loss in gravitational potential energy is equal to the sum of the elastic potential energy stored in the wire and the heat produced.
- Therefore, we can write:
\[ \Delta U = U_e + Q \]
- Rearranging gives us the heat produced \( Q \):
\[ Q = \Delta U - U_e \]
- Substituting the values we have:
\[ Q = M g l - \frac{1}{2} M g l = \frac{1}{2} M g l \]

### Summary of Results:
- Loss in gravitational potential energy: \( M g l \)
- Elastic potential energy stored in the wire: \( \frac{1}{2} M g l \)
- Heat produced: \( \frac{1}{2} M g l \)