Two springs have spring constants `k_(1)and k_(2) (k_(1)nek_(2)).` Both are extended by same force. If their elastic potential energical are `U_(1)and U_(2),` then `U_(2)` is

Options

A. `(k_(2))/(k_(1))U_(1)`
B. `(k_(1))/(k_(2))U_(1)`
C. `sqrt((k_(1))/(k_(2)))U_(1)`
D. `sqrt((k_(2))/(k_(1)))U_(1)`

Answer

To solve the problem, we need to find the relationship between the elastic potential energies \( U_1 \) and \( U_2 \) of two springs with different spring constants \( k_1 \) and \( k_2 \) when they are extended by the same force.

### Step-by-Step Solution:

1. **Understand the relationship between force and extension for springs:**
The force \( F \) exerted on a spring is given by Hooke's law:
\[ F = k \cdot x \]
where \( k \) is the spring constant and \( x \) is the extension of the spring.

2. **Set up the equations for both springs:**
For spring 1:
\[ F = k_1 \cdot x_1 \quad \text{(1)} \]
For spring 2:
\[ F = k_2 \cdot x_2 \quad \text{(2)} \]

3. **Since both springs are extended by the same force, we can equate the two expressions:**
From equations (1) and (2), we have:
\[ k_1 \cdot x_1 = k_2 \cdot x_2 \]
Rearranging gives:
\[ \frac{k_1}{k_2} = \frac{x_2}{x_1} \quad \text{(3)} \]

4. **Write the expressions for the elastic potential energy of both springs:**
The elastic potential energy \( U \) stored in a spring is given by:
\[ U = \frac{1}{2} k x^2 \]
For spring 1:
\[ U_1 = \frac{1}{2} k_1 x_1^2 \quad \text{(4)} \]
For spring 2:
\[ U_2 = \frac{1}{2} k_2 x_2^2 \quad \text{(5)} \]

5. **Divide the equations for potential energies:**
To find the ratio of \( U_1 \) to \( U_2 \):
\[ \frac{U_1}{U_2} = \frac{\frac{1}{2} k_1 x_1^2}{\frac{1}{2} k_2 x_2^2} = \frac{k_1 x_1^2}{k_2 x_2^2} \]

6. **Substituting \( \frac{x_2}{x_1} \) from equation (3):**
We can substitute \( x_2 = \frac{k_1}{k_2} x_1 \) into the ratio:
\[ \frac{U_1}{U_2} = \frac{k_1 x_1^2}{k_2 \left(\frac{k_1}{k_2} x_1\right)^2} = \frac{k_1 x_1^2}{k_2 \cdot \frac{k_1^2}{k_2^2} x_1^2} \]
Simplifying gives:
\[ \frac{U_1}{U_2} = \frac{k_1 k_2^2}{k_2 k_1^2} = \frac{k_2}{k_1} \]

7. **Rearranging gives the expression for \( U_2 \):**
\[ U_2 = U_1 \cdot \frac{k_1}{k_2} \]

### Final Result:
Thus, the expression for \( U_2 \) in terms of \( U_1 \) is:
\[ U_2 = U_1 \cdot \frac{k_1}{k_2} \]