A spring of force constant k extends by a length X on loading . If T is the tension in the spring then the energy stored in the spring is

Options

A. `T^2/(2k)`
B. `T^2/(2k^(2))`
C. `(2k)/(T^2)`
D. `(2T^(2))/k`

Answer

To find the energy stored in a spring when it is extended by a length \( X \) under a tension \( T \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Spring Force**:
The force exerted by a spring when it is stretched or compressed is given by Hooke's Law:
\[ F = kx \]
where \( F \) is the force exerted by the spring, \( k \) is the spring constant, and \( x \) is the extension or compression of the spring.

2. **Relating Tension and Extension**:
In this case, the tension \( T \) in the spring is equal to the force exerted by the spring when it is extended by \( X \):
\[ T = kX \]

3. **Energy Stored in the Spring**:
The energy \( U \) stored in the spring when it is stretched by a distance \( X \) is given by the formula:
\[ U = \frac{1}{2} k x^2 \]

4. **Substituting for \( x \)**:
We can express \( x \) in terms of \( T \) using the relationship from step 2:
\[ X = \frac{T}{k} \]

5. **Plugging \( X \) into the Energy Formula**:
Now, substituting \( X \) into the energy formula:
\[ U = \frac{1}{2} k \left(\frac{T}{k}\right)^2 \]

6. **Simplifying the Expression**:
Simplifying the above expression:
\[ U = \frac{1}{2} k \cdot \frac{T^2}{k^2} = \frac{T^2}{2k} \]

7. **Final Result**:
Thus, the energy stored in the spring when it is extended by a length \( X \) under tension \( T \) is:
\[ U = \frac{T^2}{2k} \]