When a spring is compressed by a distance `'x'`, the potential energy stored is `U_(1)`. It is further compressed by a distance `'2x'`, the increase in potential energy is `U_(2)`. The ratio of `U_(1):U_(2)`

Options

A. `1:7`
B. `1:4`
C. `1:8`
D. `1:3`

Explanation

To solve the problem, we need to find the ratio of the potential energy stored in a spring when compressed by a distance 'x' (denoted as U₁) to the potential energy when further compressed by a distance '2x' (denoted as U₂).

### Step-by-Step Solution:

1. **Understanding the Potential Energy Formula**:
The potential energy (U) stored in a spring is given by the formula:
\[ U = \frac{1}{2} k x^2 \]
where \( k \) is the spring constant and \( x \) is the compression distance.

2. **Calculating U₁**:
When the spring is compressed by a distance \( x \):
\[ U_1 = \frac{1}{2} k x^2 \]

3. **Calculating U₂**:
When the spring is further compressed by \( 2x \), the total compression becomes \( 3x \) (since it was already compressed by \( x \)):
\[ U_{\text{final}} = \frac{1}{2} k (3x)^2 = \frac{1}{2} k (9x^2) = \frac{9}{2} k x^2 \]
The increase in potential energy \( U_2 \) is the difference between the final potential energy and the initial potential energy:
\[ U_2 = U_{\text{final}} - U_1 = \frac{9}{2} k x^2 - \frac{1}{2} k x^2 = \frac{9}{2} k x^2 - \frac{1}{2} k x^2 = \frac{8}{2} k x^2 = 4 k x^2 \]

4. **Expressing U₂ in terms of U₁**:
We know that:
\[ U_1 = \frac{1}{2} k x^2 \]
Therefore, we can express \( U_2 \) in terms of \( U_1 \):
\[ U_2 = 8 \left(\frac{1}{2} k x^2\right) = 8 U_1 \]

5. **Finding the Ratio U₁ : U₂**:
Now we can find the ratio:
\[ \frac{U_1}{U_2} = \frac{U_1}{8 U_1} = \frac{1}{8} \]
Thus, the ratio of \( U_1 : U_2 \) is:
\[ U_1 : U_2 = 1 : 8 \]

### Final Answer:
The ratio of \( U_1 : U_2 \) is \( 1 : 8 \).

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