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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

1. 1 : 9
2. 1 : 11
3. 1 : 14
4. 1 : 6

Correct answer: 1 : 14

Solution

When the spring is cut into parts of lengths in the ratio 1:2:3, the spring constants of the parts become 6k, 3k, and 2k respectively (since spring constant is inversely proportional to length). For series combination, the equivalent spring constant is given by 1/k' = 1/(6k) + 1/(3k) + 1/(2k), which simplifies to k' = k/11. For parallel combination, the equivalent spring constant is k'' = 6k + 3k + 2k = 11k. Thus, k':k'' = 1:14.

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