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 :Option 1)$\frac{g}{30}\:$Option 2)$\: \frac{g}{5}\:$Option 3)$\: \frac{g}{10}\:$Option 4)$\: \frac{g}{20}\: \:$

Speed of wave on string -

$v= \sqrt{\frac{T}{\mu }}$

- wherein

$T=$ Tension in the string

$\mu =$ linear mass density

v = $\sqrt{\frac{T}{m}}$

60 = $\sqrt{\frac{Mg}{m}}$    given m << M

When the car accelerates with a

Resultant acceleration = $\sqrt{a^{2} + g^{2}}$

$\therefore 60.5 = \sqrt{\frac{M\sqrt{a^{2} + g^{2}}}{m}}$

$\left ( \frac{60.5}{60} \right )^{2} = \frac{\sqrt{a^{2} + g^{2}}}{g}$

$\left ( \frac{60.5}{60} \right )^{4} = \frac{{a^{2} + g^{2}}}{g}$

$a^{2} = g^{2} \times \frac{ \left ( {60.5} \right )^{4} }{60^{4}} - g^{2}$

$a = g \sqrt{\left ( \frac{60.5}{60} \right )^{4} - 1}$

a = 0.1837 g

= $\frac{g}{5.44}$

$a \approx \frac{g}{5}$

Option 1)

$\frac{g}{30}\:$

Option 2)

$\: \frac{g}{5}\:$

Option 3)

$\: \frac{g}{10}\:$

Option 4)

$\: \frac{g}{20}\: \:$

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