Two springs have force constants `k_(A)` such that `k_(B)=2k_(A)`. The four ends of the springs are stretched by the same force. If energy stored in spring `A` is `E`, then energy stored in spring `B` is

Options

A. a)`E/2`
B. b)`2E`
C. c)`E`
D. d)`4E`

Explanation

To solve the problem, we need to find the energy stored in spring B when the energy stored in spring A is given as E. The relationship between the force constants of the two springs is given as \( k_B = 2k_A \).

### Step-by-Step Solution:

1. **Understand the relationship between force and spring constant**:
The force \( F \) applied to a spring is related to its spring constant \( k \) and the displacement \( x \) from its equilibrium position by Hooke's law:
\[ F = k \cdot x \]

2. **Set up the equations for both springs**:
For spring A:
\[ F = k_A \cdot x_A \]
For spring B:
\[ F = k_B \cdot x_B \]
Since \( k_B = 2k_A \), we can write:
\[ F = 2k_A \cdot x_B \]

3. **Relate the displacements \( x_A \) and \( x_B \)**:
From the equations for the forces, we can equate the two expressions for force:
\[ k_A \cdot x_A = 2k_A \cdot x_B \]
Dividing both sides by \( k_A \) (assuming \( k_A \neq 0 \)):
\[ x_A = 2x_B \]
This shows that the displacement of spring A is twice that of spring B.

4. **Write the energy stored in each spring**:
The energy stored in a spring is given by the formula:
\[ E = \frac{1}{2} k x^2 \]
For spring A:
\[ E_A = \frac{1}{2} k_A x_A^2 \]
For spring B:
\[ E_B = \frac{1}{2} k_B x_B^2 \]

5. **Substitute \( x_A \) in terms of \( x_B \)**:
Since \( x_A = 2x_B \), substitute this into the energy equation for spring A:
\[ E_A = \frac{1}{2} k_A (2x_B)^2 = \frac{1}{2} k_A \cdot 4x_B^2 = 2 k_A x_B^2 \]

6. **Substitute \( k_B \) in the energy equation for spring B**:
Now substituting \( k_B = 2k_A \) into the energy equation for spring B:
\[ E_B = \frac{1}{2} (2k_A) x_B^2 = k_A x_B^2 \]

7. **Relate \( E_B \) to \( E_A \)**:
Now we have:
\[ E_A = 2 k_A x_B^2 \]
\[ E_B = k_A x_B^2 \]
Therefore, we can express \( E_B \) in terms of \( E_A \):
\[ E_B = \frac{1}{2} E_A \]

8. **Substituting \( E_A \) with \( E \)**:
Given that \( E_A = E \), we find:
\[ E_B = \frac{1}{2} E \]

### Final Answer:
The energy stored in spring B is:
\[ E_B = \frac{E}{2} \]