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 increase to $60.5 {ms}^{- 1}$ . The value of $a$ , in terms of gravitational acceleration $g$ , is close to

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$\frac{g}{20.47}$

$\frac{g}{30.47}$

$\frac{g}{10.47}$

$\frac{g}{5.47}$

answer is A.

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Detailed Solution

Velocity of transverse wave $V = \sqrt{\frac{T}{μ}}$
T = Tension in string
$μ$ = linear mass density
At the time when car is at rest
$60 = \sqrt{\frac{m g}{μ}}$
In case when car is accelerating
$60.5 = \sqrt{\frac{m {(g^{2} + a^{2})}^{1 / 2}}{μ}} \Rightarrow \frac{60.5}{60} = \sqrt{\sqrt{\frac{g^{2} + a^{2}}{g}}} {(1 + \frac{0.5}{60})}^{4} = \frac{g^{2} + a^{2}}{g^{2}} = 1 + \frac{2}{60} = g^{2} + a^{2} = g^{2} + g^{2} \times \frac{2}{60} a = g \sqrt{\frac{2}{60}} = \frac{g}{\sqrt{30}} = \frac{g}{5.47} ≃ \frac{g}{5}$