Question 91 of 223

A heavy ball of mass $M$ is suspended from the ceiling of a car by a light string of mass $m\left(m≪M\right).$ When the car is at rest, the speed of transverse waves in the string is $60{ms}^{-1}.$ When the car has acceleration $a,$ the wave-speed increases to $60.5{ms}^{-1}.$ The value of $a,$ in terms of gravitational acceleration $g,$ is closed to

A$\dfrac{g}{10}$B$\dfrac{g}{20}$C$\dfrac{g}{5}$D$\dfrac{g}{30}$

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Answer: (C) $\dfrac{g}{5}$

For standing car, $v=\sqrt{\dfrac{T}{\mu }}=\sqrt{\dfrac{Mg}{\mu }}$



For accelerating car,

$υ'=\sqrt{\dfrac{\sqrt{{(Mg)}^{2}+{(Ma)}^{2}}}{\mu }}$

$\Rightarrow \dfrac{60.5}{60}=\sqrt{\dfrac{\sqrt{{(Mg)}^{2}+{(Ma)}^{2}}}{Mg}}$

$\Rightarrow {\left(1+\dfrac{1}{120}\right)}^{2}=\dfrac{\sqrt{{(Mg)}^{2}+{(Ma)}^{2}}}{Mg}$

$\Rightarrow 1+\dfrac{2}{120}=\sqrt{1+\dfrac{{a}^{2}}{{g}^{2}}}$

$\approx \dfrac{2}{120}=\dfrac{1}{2}\dfrac{{a}^{2}}{{g}^{2}}$

$∴a=g\sqrt{\dfrac{4}{120}}=\dfrac{g}{\sqrt{30}}\approx \dfrac{g}{5}$