Force constants `K_(1) and K_(2)` of two springs are in the ratio `5:4`. They are stretched by same length. If potential energy stored in one spring is 25 J then potential energy stored in second spring is

Options

A. 25 J
B. 16 J
C. 100 J
D. 20 J

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the relationship between potential energy and spring constant
The 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.

### Step 2: Set up the ratio of the spring constants
We are given that the force constants \( K_1 \) and \( K_2 \) of the two springs are in the ratio \( 5:4 \). Therefore, we can express this as:
\[ \frac{K_1}{K_2} = \frac{5}{4} \]

### Step 3: Write the potential energy expressions for both springs
Let the potential energy stored in the first spring (with spring constant \( K_1 \)) be \( U_1 \) and in the second spring (with spring constant \( K_2 \)) be \( U_2 \). Since both springs are stretched by the same length \( x \), we can write:
\[ U_1 = \frac{1}{2} K_1 x^2 \]
\[ U_2 = \frac{1}{2} K_2 x^2 \]

### Step 4: Relate the potential energies using the ratio of spring constants
From the ratio of the spring constants, we can express \( U_1 \) in terms of \( U_2 \):
\[ \frac{U_1}{U_2} = \frac{\frac{1}{2} K_1 x^2}{\frac{1}{2} K_2 x^2} = \frac{K_1}{K_2} = \frac{5}{4} \]

### Step 5: Substitute the known value of potential energy
We know that \( U_1 = 25 \) J. Using the ratio we derived:
\[ \frac{25}{U_2} = \frac{5}{4} \]

### Step 6: Solve for \( U_2 \)
Cross-multiplying gives:
\[ 25 \cdot 4 = 5 \cdot U_2 \]
\[ 100 = 5 U_2 \]
\[ U_2 = \frac{100}{5} = 20 \text{ J} \]

### Final Answer
The potential energy stored in the second spring is \( 20 \) J.

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