The force constants of two springs are`K_(1) and K_(2)`. Both are stretched till their elastic energies are equal. If the stretching forces are ` F_(1) and F_(2) then F_(1) : F_(2)` is

Options

A. `K_(1) : K_(2)`
B. ` K_(2) : K_(1)`
C. `sqrt(k_(1)) : sqrt(K_(2))`
D. ` K_(1)^(2) : K_(2)^(2)`

Answer

To solve the problem, we need to find the ratio of the forces \( F_1 \) and \( F_2 \) applied to two springs with spring constants \( K_1 \) and \( K_2 \) respectively, given that their elastic energies are equal.

### Step-by-step Solution:

1. **Understanding the Elastic Energy**:
The elastic 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 extension of the spring.

2. **Setting Up the Equations**:
For the two springs, we have:
- For spring 1:
\[ U_1 = \frac{1}{2} K_1 x_1^2 \]
- For spring 2:
\[ U_2 = \frac{1}{2} K_2 x_2^2 \]
Since the elastic energies are equal, we can set these two equations equal to each other:
\[ \frac{1}{2} K_1 x_1^2 = \frac{1}{2} K_2 x_2^2 \]

3. **Simplifying the Equation**:
We can cancel \( \frac{1}{2} \) from both sides:
\[ K_1 x_1^2 = K_2 x_2^2 \]

4. **Relating Forces to Extensions**:
The force applied to stretch each spring is given by Hooke's Law:
- For spring 1:
\[ F_1 = K_1 x_1 \]
- For spring 2:
\[ F_2 = K_2 x_2 \]

5. **Expressing Extensions in Terms of Forces**:
We can express \( x_1 \) and \( x_2 \) in terms of the forces:
\[ x_1 = \frac{F_1}{K_1} \quad \text{and} \quad x_2 = \frac{F_2}{K_2} \]

6. **Substituting Back into the Energy Equation**:
Substitute \( x_1 \) and \( x_2 \) into the energy equation:
\[ K_1 \left(\frac{F_1}{K_1}\right)^2 = K_2 \left(\frac{F_2}{K_2}\right)^2 \]
This simplifies to:
\[ \frac{F_1^2}{K_1} = \frac{F_2^2}{K_2} \]

7. **Finding the Ratio of Forces**:
Rearranging gives:
\[ \frac{F_1^2}{F_2^2} = \frac{K_1}{K_2} \]
Taking the square root of both sides:
\[ \frac{F_1}{F_2} = \sqrt{\frac{K_1}{K_2}} \]

### Final Result:
Thus, the ratio of the forces \( F_1 : F_2 \) is:
\[ F_1 : F_2 = \sqrt{K_1 : K_2} \]