Question

Two simple pendulums of length $$0.5\,m$$  and $$2.0\,m$$  respectively are given small linear displacement in one direction at the same time. They will again be in the same phase when the pendulum of shorter length has completed oscillations

A. 5
B. 1
C. 2  
D. 3
Answer :   2
Solution :
For the pendulum to be again in the same phase, there should be difference of one complete oscillation.
If smaller pendulum completes $$n$$ oscillations the larger pendulum will complete $$\left( {n - 1} \right)$$  oscillations, so Time period of $$n$$ oscillations of first = Time period of $$\left( {n - 1} \right)$$  oscillations of second
$$\eqalign{ & {\text{i}}{\text{.e}}{\text{.}}\,\,n{T_1} = \left( {n - 1} \right){T_2} \cr & {\text{or}}\,\,n2\pi \sqrt {\frac{{{l_1}}}{g}} = \left( {n - 1} \right)2\pi \sqrt {\frac{{{l_2}}}{g}} \cr & {\text{or}}\,\,n\sqrt {{l_1}} = \left( {n - 1} \right)\sqrt {{l_2}} \cr & {\text{or}}\,\,\frac{n}{{n - 1}} = \sqrt {\frac{{{l_2}}}{{{l_1}}}} = \sqrt {\frac{{2.0}}{{0.5}}} \cr & {\text{or}}\,\,\frac{n}{{n - 1}} = 2 \cr & {\text{or}}\,\,n = 2\,n - 2 \cr & \therefore n = 2 \cr} $$

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

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Simple Harmonic Motion (SHM)

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