Two springs of spring constants 1000 N/m and 2000 N/m respectively, are stretched with the same force. The ratio of their potential energies will be

Options

A. `1 : 2`
B. `1 : 3`
C. `2 : 1`
D. `3 : 1`

Explanation

To solve the problem of finding the ratio of potential energies of two springs with different spring constants when stretched by the same force, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given values:**
- Spring constant of the first spring, \( k_1 = 1000 \, \text{N/m} \)
- Spring constant of the second spring, \( k_2 = 2000 \, \text{N/m} \)

2. **Understand the formula for potential energy in a spring:**
The potential 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.

3. **Relate force to extension:**
When a force \( F \) is applied to a spring, the extension \( x \) can be expressed as:
\[ F = k x \quad \Rightarrow \quad x = \frac{F}{k} \]

4. **Substitute \( x \) into the potential energy formula:**
We can substitute \( x \) in the potential energy formula:
\[ E = \frac{1}{2} k \left( \frac{F}{k} \right)^2 \]
Simplifying this gives:
\[ E = \frac{1}{2} k \cdot \frac{F^2}{k^2} = \frac{1}{2} \frac{F^2}{k} \]

5. **Calculate the potential energies for both springs:**
- For the first spring:
\[ E_1 = \frac{1}{2} \frac{F^2}{k_1} = \frac{1}{2} \frac{F^2}{1000} \]
- For the second spring:
\[ E_2 = \frac{1}{2} \frac{F^2}{k_2} = \frac{1}{2} \frac{F^2}{2000} \]

6. **Find the ratio of potential energies:**
The ratio of the potential energies \( \frac{E_1}{E_2} \) is:
\[ \frac{E_1}{E_2} = \frac{\frac{1}{2} \frac{F^2}{1000}}{\frac{1}{2} \frac{F^2}{2000}} = \frac{2000}{1000} = 2 \]

### Final Result:
The ratio of the potential energies of the two springs is:
\[ \frac{E_1}{E_2} = 2:1 \]