Question

Three springs having spring constants \( k \), \( 2k \), and \( 3k \) are connected (i) in series and (ii) in parallel. Let the effective spring constants for the series and parallel combinations be \( k_s \) and \( k_p \) respectively. What is the ratio \( \frac{k_p}{k_s} \)?

• A. 11:1
• B. 1:1
• C. 6:11
• D. 1:11

Hint:

For springs connected in series, the effective spring constant decreases, while for springs in parallel, the effective spring constant increases. Use the reciprocal rule for series and simple addition for parallel combinations.

Answer 1

The Correct Option is C

We are given three springs with spring constants \( k \), \( 2k \), and \( 3k \). 1. Springs in Series: When springs are connected in series, the effective spring constant \( k_s \) is given by the reciprocal of the sum of the reciprocals of the individual spring constants: \[ \frac{1}{k_s} = \frac{1}{k} + \frac{1}{2k} + \frac{1}{3k} \] Simplifying: \[ \frac{1}{k_s} = \frac{1}{k} \left( 1 + \frac{1}{2} + \frac{1}{3} \right) \] \[ \frac{1}{k_s} = \frac{1}{k} \times \frac{11}{6} \] \[ k_s = \frac{6k}{11} \] 2. Springs in Parallel: When springs are connected in parallel, the effective spring constant \( k_p \) is the sum of the individual spring constants: \[ k_p = k + 2k + 3k = 6k \] 3. Ratio \( \frac{k_p}{k_s} \): Now, the ratio of the effective spring constants is: \[ \frac{k_p}{k_s} = \frac{6k}{\frac{6k}{11}} = 11 \] Thus, the ratio \( \frac{k_p}{k_s} = 6:11 \).