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′. If they are connected in parallel and the force constant is k″, then k′ : k″ is:
Options

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′. If they are connected in parallel and the force constant is k″, then k′ : k″ is:

Options

G1 Physics · Accepted Answer

Step 1: Spring stiffness is inversely proportional to its length. Step 2: The three pieces of length fractions 1/6, 2/6 and 3/6 have constants 6k, 3k and 2k. Step 3: Series: 1/k′ = 1/(6k) + 1/(3k) + 1/(2k) = 1/k, so k′ = k. Step 4: Parallel: k″ = 6k + 3k + 2k = 11k. Step 5: Therefore k′ : k″ = 1 : 11. Final answer: (C) 1 : 11.