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 \( L_1 = nL_2 \) (n is an integer). The force constant of the spring of length ' \( L_2 \) ' is

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

Hint:

- Shorter spring $\Rightarrow$ larger spring constant - $k \propto \frac{1}{L}$

Answer 1

The Correct Option is A

Concept: Spring constant is inversely proportional to length: \[ k \propto \frac{1}{L} \]

Step 1: Given relation.
\[ L_1 = nL_2 \Rightarrow L = L_1 + L_2 = nL_2 + L_2 = (n+1)L_2 \]

Step 2: Use proportionality.
\[ \frac{K_2}{K} = \frac{L}{L_2} = \frac{(n+1)L_2}{L_2} = (n+1) \]

Step 3: Find force constant.
\[ K_2 = K(n+1) \] Answer: \( K(1 + n) \)