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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:

1:9

1:14

1:6

1:11

Solution:

spring in series
1/k' = 1/k + 1/2k + 1/3k → k' = 6k/11

parallel
k'' = k + 2k + 3k = 6k
k'/k'' = (6k/11)/(6k) = 1/11