The potential energy of a long spring when stretched by 2 cm is U. If the spring is stretched by 8 cm the potential energy stored in it is:-

Options

A. 4U
B. 8U
C. 16U
D. `U/4`

Explanation

To solve the problem, we need to determine the potential energy stored in a spring when it is stretched by 8 cm, given that the potential energy when stretched by 2 cm is U.

### Step-by-Step Solution:

1. **Understanding the Formula for Potential Energy in a Spring:**
The potential energy (U) stored in a spring when it is stretched or compressed by a distance \( X \) is given by the formula:
\[ U = \frac{1}{2} k X^2 \]
where \( k \) is the spring constant and \( X \) is the displacement from the natural length.

2. **Calculate Potential Energy for 2 cm Stretch:**
Given that when the spring is stretched by 2 cm, the potential energy is \( U \):
\[ U = \frac{1}{2} k (2)^2 \]
Simplifying this, we get:
\[ U = \frac{1}{2} k \cdot 4 = 2k \]
Thus, we can express \( k \) in terms of \( U \):
\[ k = \frac{U}{2} \]

3. **Calculate Potential Energy for 8 cm Stretch:**
Now, we need to find the potential energy when the spring is stretched by 8 cm:
\[ U' = \frac{1}{2} k (8)^2 \]
Substituting \( k \) from the previous step:
\[ U' = \frac{1}{2} \left(\frac{U}{2}\right) (8)^2 \]
Simplifying this, we get:
\[ U' = \frac{1}{2} \cdot \frac{U}{2} \cdot 64 = \frac{U \cdot 64}{4} = 16U \]

4. **Conclusion:**
Therefore, the potential energy stored in the spring when stretched by 8 cm is:
\[ U' = 16U \]

### Final Answer:
The potential energy stored in the spring when stretched by 8 cm is \( 16U \).

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