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 1: Understand the formula for energy stored in a spring
The energy \( E \) stored in a spring is given by the formula:
\[ E = \frac{1}{2} k x^2 \]
where \( k \) is the spring constant and \( x \) is the extension of the spring.

### Step 2: Relate tension to extension
According to Hooke's Law, the tension \( T \) in the spring is directly proportional to the extension \( x \):
\[ T = kx \]
From this, we can express the extension \( x \) in terms of tension \( T \):
\[ x = \frac{T}{k} \]

### Step 3: Substitute \( x \) in the energy formula
Now, we can substitute the expression for \( x \) into the energy formula:
\[ E = \frac{1}{2} k \left(\frac{T}{k}\right)^2 \]
This simplifies to:
\[ E = \frac{1}{2} k \cdot \frac{T^2}{k^2} \]
\[ E = \frac{T^2}{2k} \]

### Conclusion
Thus, the energy stored in the spring when it is extended by a length \( X \) under a tension \( T \) is:
\[ E = \frac{T^2}{2k} \]