First slide

Simple hormonic motion

Question

A rectangular block of mass m and area of cross-section A floats in a liquid of density $ρ$ . If it is given a small vertical displacement from equilibrium it undergoes oscillation with a time period T then:

Moderate

A

$T \propto \sqrt{ρ}$
B

$T \propto \frac{1}{\sqrt{A}}$
C

$T \propto \frac{1}{ρ}$
D

$T \propto \frac{1}{\sqrt{m}}$

Solution

Force applied on the body will be equal to upthrust for vertical oscillations.
Let block is displaced through x m, then the weight of displaced water or upthrust (upwards)
= _-Ax $ρ$ g
Where A is area of cross-section of the block and $ρ$ is its density. This must be equal to force (= ma) applied, where m is mass of the block and a is acceleration.
$∴$ ma = _ Ax $ρ$ g
or $a = - \frac{Aρg}{m} x = - ω^{2} x$
This is the equation of simple harmonic motion.
Time period of oscillation
$T = \frac{2 π}{ω} = 2 π \sqrt{\frac{m}{Aρg}} \Rightarrow T \propto \frac{1}{\sqrt{A}}$

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