Question

A string is stretched between fixed points separated by $$75.0\,cm.$$  It is observed to have resonant frequencies of $$420\,Hz$$  and $$315\,Hz.$$  There are no other resonant frequencies between these two. The lowest resonant frequency for this strings is

A. $$155\,Hz$$
B. $$205\,Hz$$
C. $$10.5\,Hz$$
D. $$105\,Hz$$  
Answer :   $$105\,Hz$$
Solution :
Given $$l = 75\,cm,{f_1} = 420\,Hz\,{\text{and}}\,{f_2} = 315\,Hz.$$
As, two consecutive resonant frequencies for a string fixed at both ends will be
$$\eqalign{ & {f_1} = \frac{{nv}}{{2l}}\,\,{\text{and}}\,\,{f_2} = \frac{{\left( {n + 1} \right)v}}{{2l}} \cr & \Rightarrow {f_2} - {f_1} = 420 - 315 \cr & \Rightarrow \frac{{\left( {n + 1} \right)v}}{{2l}} - \frac{{nv}}{{2l}} = 105\,Hz \cr & \Rightarrow \frac{v}{{2l}} = 105\,Hz \cr} $$
Thus, lowest resonant frequency of a string is $$105\,Hz.$$

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. $$2\pi \sqrt {\frac{L}{{g\tan \alpha }}} $$
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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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