A heavy ball of mass $M$ is suspended from the ceiling of a car by a light string of mass $m$ ($m \ll M$). When the car is at rest, the speed of transverse waves in the string is $60\ \text{m s}^{-1}$. When the car has acceleration $a$, the wave-speed increases to $60.5\ \text{m s}^{-1}$. The value of $a$, in terms of gravitational acceleration $g$, is close to
Answer: (C) $\dfrac{g}{5}$
The wave speed is $v = \sqrt{T/\mu}$. At rest $T = Mg$; when the car accelerates, the string tilts and $T = M\sqrt{g^2 + a^2}$. Hence
$$\frac{v'}{v} = \left(\frac{g^2+a^2}{g^2}\right)^{1/4} = \frac{60.5}{60} = 1 + \frac{1}{120}$$
Using the binomial approximation,
$$1 + \frac{a^2}{4g^2} \approx 1 + \frac{1}{120} \Rightarrow \frac{a^2}{g^2} = \frac{1}{30} \Rightarrow a \approx 0.18g \approx \frac g5$$
Solution by Sreeraj P, M.Sc Physics