Figure below shows a shampoo bottle in a perfect cylindrical shap… - Vidyadip

MCQ

Figure below shows a shampoo bottle in a perfect cylindrical shape. In a simple experiment, the stability of the bottle filled with different amount of shampoo volume is observed. The bottle is tilted from one side and then released. Let the angle $\theta$ depicts the critical angular displacement resulting, in the bottle losing its stability and tipping over. Choose the graph correctly depicting the fraction $f$ of shampoo filled $(f=1$ corresponds to completely filled) versus the tipping angle $\theta$

A
B
C
✓

✓

Answer

Correct option: D.

d
(d)

The situation can be depicted as

where, $h_b=$ height of bottle,

$h_s=$ height of shampoo

and $R=$ radius of bottle.

For critical angular displacement, $m g$ would pass through tilted side as shown above.

From the above figure,

$\tan \theta=\frac{R}{h_s}$

Also, fraction of shampoo, $f=\frac{h_s}{h_b}$

$\Rightarrow h_s=f h_b$

$\tan \theta=\frac{R}{f h_b}$

or

$\theta=\tan ^{-1}\left(\frac{R}{f h_b}\right)$

$\therefore$ If $f$ increases, then $\theta$ will increase upto $0.5 f$ and afterward it decreases. It is correctly shown in graph $(d)$.

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