Two springs of spring constants `1000 N m^(-1)` and `2000 N m^(-1)` are stretched with same force. They will have potential energy in the ratio of

Options

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

Explanation

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

### Step 1: Understand the relationship between force, spring constant, and displacement
The force \( F \) applied to a spring is related to the spring constant \( k \) and the displacement \( x \) from its equilibrium position by Hooke's Law:
\[ F = kx \]

### Step 2: Express displacement in terms of force and spring constant
From Hooke's Law, we can rearrange the equation to solve for displacement \( x \):
\[ x = \frac{F}{k} \]

### Step 3: Write the formula for potential energy stored in a spring
The potential energy \( U \) stored in a spring when it is stretched or compressed is given by the formula:
\[ U = \frac{1}{2} k x^2 \]

### Step 4: Substitute the expression for displacement into the potential energy formula
Substituting \( x = \frac{F}{k} \) into the potential energy formula, we get:
\[ U = \frac{1}{2} k \left(\frac{F}{k}\right)^2 \]
This simplifies to:
\[ U = \frac{1}{2} k \cdot \frac{F^2}{k^2} = \frac{F^2}{2k} \]

### Step 5: Analyze the potential energy for both springs
Let’s denote the spring constants as \( k_1 = 1000 \, \text{N/m} \) and \( k_2 = 2000 \, \text{N/m} \).
The potential energy for spring 1 is:
\[ U_1 = \frac{F^2}{2k_1} \]
And for spring 2:
\[ U_2 = \frac{F^2}{2k_2} \]

### Step 6: Find the ratio of potential energies
Now, we can find the ratio of the potential energies \( U_1 \) and \( U_2 \):
\[ \frac{U_1}{U_2} = \frac{\frac{F^2}{2k_1}}{\frac{F^2}{2k_2}} = \frac{k_2}{k_1} \]
Substituting the values of \( k_1 \) and \( k_2 \):
\[ \frac{U_1}{U_2} = \frac{2000}{1000} = 2 \]

### Conclusion
Thus, the ratio of potential energies \( U_1 : U_2 \) is:
\[ U_1 : U_2 = 2 : 1 \]