You are holding a shallow circular container of radius R, filled … - Vidyadip

MCQ

You are holding a shallow circular container of radius $R$, filled with water to a height $h ( h < < R )$. When yon walk with speed $v$, it is seen that water starts spilling over. This happens due to the resonance of the periodic impulse given to the container (due to walking) with the oscillation of the water in the container. If the time period of water oscillating in the container is inversely proportional to $\sqrt{ h }$, then $v$ is proportional to

A
$R$
B
$\sqrt{R}$
C
$1 / \sqrt{R}$
✓
$1 / R$

✓

Answer

Correct option: D.

$1 / R$

d
(D)

At resonance, frequency of walking $=$ Frequency of oscillation

$f _{\text {milking }}=\frac{ v }{\ell}$ (where $\ell$ is length of step \& $v$ is velocity)

$\Rightarrow \frac{ v }{\ell}= f _{ osc. }$

$f _{\text {osc. }}= kh ^5 R ^y g ^2( k$ is dimension less constant)

By question,

$x =\frac{1}{2} \Rightarrow T ^{-1}= L ^{\frac{1}{2}} L ^v\left( LT ^{-2}\right)^{z}$

$\Rightarrow z =\frac{1}{2}, y =-1$

$\Rightarrow \frac{ v }{\ell}= k \sqrt{ h } R ^{-1} g ^{\frac{1}{2}} \Rightarrow v \propto \frac{1}{ R }$

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