Question

A spring of force constant $k$ is cut into lengths of ratio $1 : 2 : 3$. 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. 1 : 9
• B. 1 : 11
• C. 1 : 14
• D. 1 : 16

Answer 1

The Correct Option is B

 To solve the problem, we need to analyze the behavior of springs when they are cut and then connected in different configurations.

Step 1: Understanding the Force Constant of a Spring

For a spring with original force constant \(k\), if it is cut into smaller lengths, each shorter spring will have a different force constant. The force constant of a spring is inversely proportional to its length.

Step 2: Force Constant of the Individual Sections

The spring is cut into three segments with a length ratio 1:2:3. Let the segments have lengths \(L_1 = L, L_2 = 2L, L_3 = 3L\).

The force constant of each section can be given as:

• Force constant of the first spring segment: \(k_1 = 6k\)
• Force constant of the second spring segment: \(k_2 = 3k\)
• Force constant of the third spring segment: \(k_3 = 2k\)

This is because the original length is divided as 1:2:3, so the force constant would be proportional to the inverse of these lengths.

Step 3: Springs Connected in Series

When springs are connected in series, the reciprocal of the equivalent spring constant \(k\) is the sum of the reciprocals of the individual spring constants:

\(\frac{1}{k'} = \frac{1}{k_1} + \frac{1}{k_2} + \frac{1}{k_3} = \frac{1}{6k} + \frac{1}{3k} + \frac{1}{2k}\)

Calculating this, we have:

\(\frac{1}{k'} = \frac{1}{6k} + \frac{2}{6k} + \frac{3}{6k} = \frac{1}{k}\)

Thus, \(k' = k\).

Step 4: Springs Connected in Parallel

When springs are connected in parallel, the equivalent force constant \(k'\) is the sum of their individual force constants:

\(k'' = k_1 + k_2 + k_3 = 6k + 3k + 2k = 11k\)

Step 5: Ratio of the Force Constants

Now, we need to find the ratio \(k' : k''\):

\(k' : k'' = k : 11k = 1 : 11\)

Conclusion: The ratio \(k' : k''\) is 1 : 11. Therefore, the correct answer is:

Option	Ratio
1 : 9	Incorrect
1 : 11	Correct
1 : 14	Incorrect
1 : 16	Incorrect