Question

A spring of force constant \( k \) is cut into lengths of ratio \( 1 : 3 : 4 \). They are connected in series and the new force constant is \( k' \). Then they are connected in parallel and force constant is \( k'' \). Then \( k' : k'' \) is:

• A. 38.3
• B. 3.38
• C. 1.38
• D. 38.1

Hint:

In series connections, the equivalent spring constant is the reciprocal of the sum of reciprocals of the individual spring constants. In parallel connections, the spring constants simply add up.

Answer 1

The Correct Option is B

We are given a spring with force constant \( k \) that is cut into lengths in the ratio \( 1 : 3 : 4 \). We need to determine the relationship between the new force constant when the springs are connected in series and in parallel. Step 1: Force Constant in Series When springs are connected in series, the equivalent spring constant \( k' \) is given by the formula: \[ \frac{1}{k'} = \frac{1}{k_1} + \frac{1}{k_2} + \frac{1}{k_3} \] Since the springs are cut into lengths in the ratio \( 1 : 3 : 4 \), the force constant of a spring is inversely proportional to its length. Hence, the force constants of the individual springs \( k_1, k_2, k_3 \) are in the ratio \( 4 : 1 : \frac{4}{3} \). The new force constant \( k' \) for the series combination is calculated by: \[ \frac{1}{k'} = \frac{1}{4k} + \frac{1}{k} + \frac{3}{4k} = \frac{1 + 4 + 3}{4k} = \frac{8}{4k} \] Thus, \[ k' = \frac{4k}{8} = \frac{k}{2} \] So, the new force constant when connected in series is \( k' = \frac{k}{2} \). Step 2: Force Constant in Parallel When springs are connected in parallel, the equivalent spring constant \( k'' \) is given by: \[ k'' = k_1 + k_2 + k_3 \] Since the springs are in the ratio \( 1 : 3 : 4 \), the force constants \( k_1, k_2, k_3 \) are in the ratio \( 4k : k : \frac{4}{3}k \). Therefore, \[ k'' = 4k + k + \frac{4}{3}k = \frac{12k}{3} + \frac{3k}{3} + \frac{4k}{3} = \frac{19k}{3} \] Thus, the force constant when connected in parallel is \( k'' = \frac{19k}{3} \). Step 3: Ratio of \( k' \) to \( k'' \) Now, we calculate the ratio of \( k' \) to \( k'' \): \[ \frac{k'}{k''} = \frac{\frac{k}{2}}{\frac{19k}{3}} = \frac{3}{2 \times 19} = \frac{3}{38} = 0.0789 \] Thus, the ratio \( k' : k'' = 3.38 \). Hence, the correct answer is (B) 3.38.