First slide

Fluid statics

Question

A wooden cylinder is floating in water with its axis vertical and its submerged length is 10 cm. If it is slightly pushed into water and released, its starts oscillating with time period T. If g = 10 m/s² find T in second.

Moderate

A

$\frac{π}{5}$
B

$\frac{2π}{5}$
C

$\frac{π}{10}$
D

$\frac{π}{2}$

Solution

When the cylinder is in equilibrium, $mg= ρ .A.l.g$
when the submerged length is (l + x), net upward force acting on the cylinder is equal to $ρA (l + x) g - mg = (ρAlg - mg) + ρAg . x = ρAgx$
$∴ ma = - ρAgx \Rightarrow ma = - \frac{m}{l} g . x \Rightarrow a = - (\frac{g}{l}) x$
$∴ ω = \sqrt{\frac{g}{l}} \Rightarrow T = \frac{2 π}{ω} = 2 π \sqrt{\frac{l}{g}} = 2 π \sqrt{\frac{0 .1}{10}} S = \frac{π}{5} \sec$

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