Question

A spring has length \( L \) and force constant \( K \). It is cut into two springs of length \( L_1 \) and \( L_2 \) such that \( L_1 = N L_2 \) (N is an integer). The force constant of the spring of length \( L_1 \) is

• A. \( \frac{K(N + 1)}{N} \)
• B. \( \frac{K(N + 1)}{2N} \)
• C. \( \frac{K(N - 1)}{N} \)
• D. \( \frac{K(N - 1)}{2N} \)

Hint:

When a spring is cut into parts, the force constant of each part is inversely proportional to its length.

Answer 1

The Correct Option is A

Step 1: Understanding the spring constant.
The force constant of a spring, \( k \), is related to the length of the spring and its force constant by the relation: \[ k = \frac{K}{L} \] where \( K \) is the force constant of the original spring and \( L \) is its length. Step 2: Use the relation for two springs in series.
When the spring is cut into two parts of lengths \( L_1 \) and \( L_2 \), and their force constants are \( k_1 \) and \( k_2 \) respectively, the force constants are inversely proportional to the lengths: \[ k_1 = \frac{K}{L_1}, \quad k_2 = \frac{K}{L_2} \] Given that \( L_1 = N L_2 \), the force constant of the spring of length \( L_1 \) will be: \[ k_1 = \frac{K}{L_1} = \frac{K}{N L_2} \] Substituting the relation between \( L_1 \) and \( L_2 \), we get: \[ k_1 = \frac{K(N + 1)}{N} \] Step 3: Conclusion.
Thus, the force constant of the spring of length \( L_1 \) is \( \frac{K(N + 1)}{N} \), which corresponds to option (A).