Question

A $$U$$-tube is of non uniform cross-section. The area of cross-sections of two sides of tube are $$A$$ and $$2A$$  (see fig.). It contains non-viscous liquid of mass $$m.$$ The liquid is displaced slightly and free to oscillate. Its time period of oscillations is
Simple Harmonic Motion (SHM) mcq question image

A. $$T = 2\pi \sqrt {\frac{m}{{3\rho gA}}} $$  
B. $$T = 2\pi \sqrt {\frac{m}{{2\rho gA}}} $$
C. $$T = 2\pi \sqrt {\frac{m}{{\rho gA}}} $$
D. None of these
Answer :   $$T = 2\pi \sqrt {\frac{m}{{3\rho gA}}} $$
Solution :
Suppose the liquid in left side limb is displaced slightly by $$y,$$ the liquid in right limb will increase by $$\frac{y}{2}.$$ The restoring force
$$\eqalign{ & F = - PA = - \rho g\left( {\frac{{3y}}{2}} \right) \times 2A = 3\rho gA\left( { - y} \right). \cr & a = \frac{F}{m} = \frac{{3\rho gA\left( { - y} \right)}}{m} \cr} $$
Simple Harmonic Motion (SHM) mcq solution image
On comparing with, $$a = - {\omega ^2}y,$$   we get
$$\omega = \sqrt {\frac{{3\rho gA}}{m}} \,\,{\text{and}}\,\,T = 2\pi \sqrt {\frac{m}{{3\rho gA}}} $$

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

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Simple Harmonic Motion (SHM)

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