First slide

Applications of SHM

Question

A brass cube of side a and density $σ$ is floating in mercury of density $ρ$ . If the cube is displaced a bit vertically, it executes S.H.M. Its time period will be

Moderate

A

$2 π \sqrt{\frac{σ a}{ρ g}}$
B

$2 π \sqrt{\frac{ρ a}{σ g}}$
C

$2 π \sqrt{\frac{ρ g}{σ a}}$
D

$2 π \sqrt{\frac{σ g}{ρ a}}$

Solution

As a is the side of cube $σ$ is its density.
Mass of cube is $a^{2} σ,$ its weight $= a^{3} σg$
Let h be the height of cube immersed in liquid of density $ρ$ in equilibrium then, $F = a^{2} h ρ g = Mg = a^{3} σ g$
If it is pushed down by y then the buoyant force $F^{'} = a^{2} (h + y) ρ g$
Restoring force is $ΔF = F^{'} - F = a^{2} (h + y) σ g - a^{2} h σ g$
$= a^{2} y ρ g$
Restoring acceleration $= \frac{ΔF}{M} = - \frac{a^{2} yρ g}{M} = - \frac{a^{2} ρ g}{a^{3} σ} y$
Motion is S.H.M.
$\Rightarrow T = 2 π \sqrt{\frac{a^{3} σ}{a^{2} ρ g}} = 2 π \sqrt{\frac{aσ}{ρ g}}$

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