A cylindrical plastic bottle of negligible mass is filled with...

A cylindrical plastic bottle of negligible mass is filled with

310ml

of water and left floating in a pond with still water. If pressed downward slightly and released, it starts performing simple harmonic motion at angular frequency

\omega .

If the radius of the bottle is

2.5cm

then

\omega

is close to:

(

density of water

= {10}^{3}kg / {m}^{3} )

Options

A5.00 rad $\sec^{- 1}$
B $2.50rad{sec}^{- 1}$
C ${7.9 rad sec}^{- 1}$
D $3.75rad{sec}^{- 1}$

Correct answer

C. ${7.9 rad sec}^{- 1}$

Step-by-step solution

Let at equilibrium,

y

length of bottle is immerged in water:

So,

mg = {f}_{buo}

mg = ρ

\left(Ay\right)g

.....

( 1 )

[Equilibrium Condition]<br /><br />Now it is further pushed by distance

<br /><br />

{F}_{restoring} = ρ [ A\left(x + y )\right]g - mg

<br /><br />

m{\omega}^{2}x = ρAxg + ρAyg - mg

<br /><br />

\omega = \sqrt{\dfrac{ρAg}{m}} = \sqrt{\dfrac{ρAg}{ρV}} = \sqrt{\dfrac{\pi{r}^{2}g}{V}} =7.9 rad / sec

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