A heavy ball of mass \(M\) is suspended from the ceiling of 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

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Correct Answer

(C) \(\frac{g}{{5}}\)

Detailed explanation

\(60=\sqrt{\frac{M g}{\mu}}\)

\(60.5=\sqrt{\frac{M\left(g^{2}+a^{2}\right)^{1 / 2}}{\mu}} \Rightarrow \frac{60.5}{60}=\sqrt{\sqrt{\frac{g^{2}+a^{2}}{g^{2}}}}\)

\(\left(1+\frac{0.5}{60}\right)^{4}=\frac{g^{2}+a^{2}}{g^{2}}=1+\frac{2}{60}\)

\(\Rightarrow g^{2}+a^{2}=g^{2}+g^{2} \times \frac{2}{60}\)

\(a=g \sqrt{\frac{2}{60}}=\frac{g}{\sqrt{30}}=\frac{g}{5.47}\)

\(\simeq \frac{g}{5}\)

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