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A uniform cylinder of length L, mass M and cross-sectional area A hangs vertically from a fixed point by a massless spring of constant k, and at equilibrium it is half submerged in a liquid of density σ. Given a small downward push, it oscillates vertically with small amplitude. Its time period T is

Asked in JEE Main Online 2013 · Floating bodies and gas columns

• (1) smaller than 2π[M/(k+Aσ g)]^1/2
• (2) 2π√/Mk
• (3) larger than 2π[M/(k+Aσ g)]^1/2
• (4) 2π[M/(k+Aσ g)]^1/2

Step-by-step solution

Why the other options are wrong

• (1) The two restoring effects simply add; there is nothing left over to make the period smaller.
• (2) Ignores the buoyant restoring force altogether.
• (3) Again, the expression is exact, not a bound.

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