SATHEE: Simple Harmonic Motion 3 Question 10

Simple Harmonic Motion 3 Question 10

10. A solid sphere of radius $R$ is floating in a liquid of density $ρ$ with half of its volume submerged. If the sphere is slightly pushed and released, it starts performing simple harmonic motion. Find the frequency of these oscillations.

(2004, 4M)

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Answer:

Correct Answer: 10. $\frac{1}{2 π} \sqrt{\frac{3 g}{2 R}}$

Solution:

Half of the volume of sphere is submerged.

For equilibrium of sphere, weight $=$ upthrust

$\begin{aligned} ∴ V ρ_{s} g & = \frac{V}{2} (ρ_{L}) (g) \\ ρ_{s} & = \frac{ρ_{L}}{2} \end{aligned}$

When slightly pushed downwards by $x$ , weight will remain as it is while upthrust will increase. The increased upthrust will become the net restoring force (upwards).

$\begin{aligned} F = - (extra upthrust) \\ = - (extra volume immersed) (ρ_{L}) (g) \\ or m a = - (π R^{2}) x ρ_{L} g (a = acceleration) \\ ∴ \frac{4}{3} π R^{3} \frac{ρ}{2} a = - (π R^{2} ρ g) x \\ ∴ a = - \frac{3 g}{2 R} x \end{aligned}$

as $a \propto - x$ , motion is simple harmonic.

Frequency of oscillation, $f = \frac{1}{2 π} \sqrt{\frac{a}{x}}$

$= \frac{1}{2 π} \sqrt{\frac{3 g}{2 R}}$

Simple Harmonic Motion 3 Question 10

10. A solid sphere of radius $R$ is floating in a liquid of density $ρ$ with half of its volume submerged. If the sphere is slightly pushed and released, it starts performing simple harmonic motion. Find the frequency of these oscillations.

Answer:

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Simple Harmonic Motion 3 Question 10

10. A solid sphere of radius R is floating in a liquid of density ρ with half of its volume submerged. If the sphere is slightly pushed and released, it starts performing simple harmonic motion. Find the frequency of these oscillations.

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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10. A solid sphere of radius $R$ is floating in a liquid of density $ρ$ with half of its volume submerged. If the sphere is slightly pushed and released, it starts performing simple harmonic motion. Find the frequency of these oscillations.