A cuboidal piece of wood has dimensions a, b and c in S.I system of units. Its relative density is d which is numerically equal to 1/a. It is floating in a large body of water such that side a is vertical. it is pushed down a bit and released. The time period of SHM executed by it is ________seconds. $(Assume g = π^{2} {ms}^{- 2})$

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Detailed Solution

Consider height h is inside the water initially

$∴ h (bc) σ g = abc \cdot ρ g$

$h = \frac{ρ}{σ} (a) = d a$ ......(1)

Now it is pushed down by x

$∴$ Extra Buoyant force will act as restoring force and is given by

$F_{R} = - B_{extra} = - x (bc) σ g$

$∴$ Acceleration $a = \frac{F_{R}}{M} = \frac{F_{R}}{abc ρ}$

$= - \frac{x (b c) σ g}{a b c ρ} = - \frac{x g}{d a}$

Comparing with $A = - ω^{2} x; ω = \sqrt{\frac{g}{a \cdot d}}$

$∴$ Time period $T = 2 π \sqrt{\frac{a \cdot d}{g}}$

Here $d = \frac{1}{a}$

$∴ T = 2 π \sqrt{\frac{a \cdot \frac{1}{a}}{π^{2}}} = \frac{2 π}{π} = 2 \sec$

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