Two springs have their force constants in the ratio of `3:4`. Both the springs are stretched by applying equal force F. If elongation in first spring is x then elogation is second spring is

Options

A. `3x`
B. `4x`
C. `(4)/(3)x`
D. `(3)/(4)x`

Answer

To solve the problem, we need to analyze the relationship between the force constants of the springs and their elongations when the same force is applied.

### Step-by-Step Solution:

1. **Understanding the relationship between force, spring constant, and elongation**:
The force exerted on a spring is given by Hooke's Law, which states:
\[ F = k \cdot x \]
where \( F \) is the force applied, \( k \) is the spring constant, and \( x \) is the elongation of the spring.

2. **Setting up the problem**:
Let the spring constants of the two springs be \( k_1 \) and \( k_2 \). According to the problem, the ratio of the spring constants is:
\[ \frac{k_1}{k_2} = \frac{3}{4} \]
This means we can express \( k_1 \) and \( k_2 \) in terms of a common variable. Let:
\[ k_1 = 3k \quad \text{and} \quad k_2 = 4k \]
for some constant \( k \).

3. **Applying the same force to both springs**:
When the same force \( F \) is applied to both springs, we can write the elongations for each spring:
- For the first spring:
\[ F = k_1 \cdot x_1 \implies x_1 = \frac{F}{k_1} \]
- For the second spring:
\[ F = k_2 \cdot x_2 \implies x_2 = \frac{F}{k_2} \]

4. **Substituting the values of \( k_1 \) and \( k_2 \)**:
Now, substituting \( k_1 \) and \( k_2 \) into the equations for elongation:
- For the first spring:
\[ x_1 = \frac{F}{3k} \]
- For the second spring:
\[ x_2 = \frac{F}{4k} \]

5. **Finding the relationship between \( x_1 \) and \( x_2 \)**:
We can express \( x_2 \) in terms of \( x_1 \):
\[ x_1 = \frac{F}{3k} \quad \text{and} \quad x_2 = \frac{F}{4k} \]
To find \( x_2 \) in terms of \( x_1 \), we can take the ratio:
\[ \frac{x_1}{x_2} = \frac{\frac{F}{3k}}{\frac{F}{4k}} = \frac{4}{3} \]
Thus,
\[ x_2 = \frac{3}{4} x_1 \]

6. **Substituting \( x_1 = x \)**:
Since the elongation in the first spring is given as \( x \), we have:
\[ x_2 = \frac{3}{4} x \]

### Final Answer:
The elongation in the second spring is:
\[ x_2 = \frac{3}{4} x \]