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 :

Answers (1)

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}

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