Question

A rectangular block of mass $$m$$ and area of cross-section $$A$$ floats in a liquid of density $$\rho .$$ If it is given a small vertical displacement from equilibrium, it undergoes oscillation with a time period $$T.$$ Then

A. $$T \propto \sqrt \rho $$
B. $$T \propto \frac{1}{{\sqrt A }}$$  
C. $$T \propto \frac{1}{\rho }$$
D. $$T \propto \frac{1}{{\sqrt m }}$$
Answer :   $$T \propto \frac{1}{{\sqrt A }}$$
Solution :
Let block be displaced through $$x$$ $$m,$$ then weight of displaced water or upthrust, (upwards) is given by Archimedes principle
$${F_b} = - Ax\rho g$$
where, $$A$$ is the area of cross-section of the block and $$\rho $$ is its density. This must be equal to force $$\left( { = ma} \right)$$  applied, where, $$m$$ is the mass of the block and $$a$$ is the acceleration.
$$\therefore ma = - Ax\rho g\,\,{\text{or}}\,\,a = - \frac{{A\rho g}}{m}x = - {\omega ^2}x$$
This is the equation of simple harmonic motion. Time period of oscillation,
$$\eqalign{ & T = \frac{{2\pi }}{\omega } = 2\pi \sqrt {\frac{m}{{A\rho g}}} \cr & \Rightarrow T \propto \frac{1}{{\sqrt A }} \cr} $$

Releted MCQ Question on
Oscillation and Mechanical Waves >> Simple Harmonic Motion (SHM)

Releted Question 1

Two bodies $$M$$ and $$N$$ of equal masses are suspended from two separate massless springs of spring constants $${k_1}$$ and $${k_2}$$ respectively. If the two bodies oscillate vertically such that their maximum velocities are equal, the ratio of the amplitude of vibration of $$M$$ to that of $$N$$ is

A. $$\frac{{{k_1}}}{{{k_2}}}$$
B. $$\sqrt {\frac{{{k_1}}}{{{k_2}}}} $$
C. $$\frac{{{k_2}}}{{{k_1}}}$$
D. $$\sqrt {\frac{{{k_2}}}{{{k_1}}}} $$
View Answer
Releted Question 2

A particle free to move along the $$x$$-axis has potential energy given by $$U\left( x \right) = k\left[ {1 - \exp \left( { - {x^2}} \right)} \right]$$      for $$ - \infty \leqslant x \leqslant + \infty ,$$    where $$k$$ is a positive constant of appropriate dimensions. Then

A. at points away from the origin, the particle is in unstable equilibrium
B. for any finite nonzero value of $$x,$$ there is a force directed away from the origin
C. if its total mechanical energy is $$\frac{k}{2},$$  it has its minimum kinetic energy at the origin.
D. for small displacements from $$x = 0,$$  the motion is simple harmonic
View Answer
Releted Question 3

The period of oscillation of a simple pendulum of length $$L$$ suspended from the roof of a vehicle which moves without friction down an inclined plane of inclination $$\alpha ,$$ is given by

A. $$2\pi \sqrt {\frac{L}{{g\cos \alpha }}} $$
B. $$2\pi \sqrt {\frac{L}{{g\sin \alpha }}} $$
C. $$2\pi \sqrt {\frac{L}{g}} $$
D. $$2\pi \sqrt {\frac{L}{{g\tan \alpha }}} $$
View Answer
Releted Question 4

A particle executes simple harmonic motion between $$x = - A$$  and $$x = + A.$$  The time taken for it to go from 0 to $$\frac{A}{2}$$ is $${T_1}$$ and to go from $$\frac{A}{2}$$ to $$A$$ is $${T_2.}$$ Then

A. $${T_1} < {T_2}$$
B. $${T_1} > {T_2}$$
C. $${T_1} = {T_2}$$
D. $${T_1} = 2{T_2}$$
View Answer

Practice More Releted MCQ Question on
Simple Harmonic Motion (SHM)

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Basic PhysicsUnit and MeasurementKinematicsLaws of MotionFrictionUniform Circular MotionImpulseMomentumWork Energy and PowerRotational MotionGravitationMechanical Properties of Solids and Fluids
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Electrostatics and MagnetismElectric ChargesElectric FieldElectric PotentialCapacitors and DielectricsMagnetic Effect of CurrentMagnetic MaterialsElectromagnetic WavesElectric CurrentElectromagnetic InductionAlternating CurrentSemiconductors and Electronic Devices
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Modern PhysicsDual Nature of Matter and RadiationAtoms And NucleiAtoms or Nuclear Fission and FusionRadioactivityModern Physics MiscellaneousSemiconductors and Electronic Devices
Oscillation and Mechanical WavesSimple Harmonic Motion (SHM)Waves