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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
Explanation
Pieces have constants 6k, 3k, 2k; series gives k, parallel gives 11k.
Detailed Solution
Spring constant $\propto \frac{1}{length}$, i.e. $k \propto \frac{1}{l}$
$k_1 = 6k$, $k_2 = 3k$, $k_3 = 2k$
In series: $\frac{1}{k'} = \frac{1}{6k} + \frac{1}{3k} + \frac{1}{2k} = \frac{6}{6k} \Rightarrow k' = k$
In parallel: $k'' = 6k + 3k + 2k = 11k$
$\frac{k'}{k''} = \frac{1}{11}$, i.e. $k' : k'' = 1 : 11$