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Oscillations

JEE Main level, single correct, medium

132 of 194

Question 132 of 194

A cylindrical plastic bottle of negligible mass is filled with $310 ml$ 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 $2.5 cm$ then $\omega$ is close to: $($ density of water $={10}^{3}kg/{m}^{3})$

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Answer: (C) ${7.9 rad sec}^{-1}⁡$

Let at equilibrium,

y

length of bottle is immerged in water:

So,

mg={f}_{buo}

mg=\rho \left(Ay\right)g

.....

(1)

[Equilibrium Condition]

Now it is further pushed by distance

x

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

m{\omega }^{2}x=\rho Axg+\rho Ayg-mg

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

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JEE Main 2025 Apr 2 Shift 1 question paper

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