Q. A heavy ball of mass M is suspended from the ceiling of a car by a light string of mass m (m < < M). 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 closest to :

Solution:

$60 = \sqrt{\frac{Mg}{\mu}} $
$ 60.5 = \sqrt{\frac{M\left(g^{2} +a^{2}\right)^{1/2}}{\mu}} \Rightarrow \frac{60.5}{60} = \sqrt{\sqrt{\frac{g^{2} +a^{2}}{g^{2}}} } $
$ \left(1+ \frac{0.5}{60}\right)^{4} = \frac{g^{2} +a^{2}}{g^{2}} = 1 + \frac{2}{ 60} $
$ \Rightarrow g^{2} +a^{2} = g^{2} + g^{2} \times\frac{2}{60} $
$ a =g \sqrt{\frac{2}{60}} = \frac{g}{\sqrt{30}} = \frac{g}{5.47} $
$ \simeq \frac{g}{5} $

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