Question

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 $60m/s$. When the car has acceleration 𝑎, the wave-speed increases to $60.5m/s$. The value of 𝑎, in terms of gravitational acceleration $g$, is closest to

• A. $\frac{g}{20}$
• B. $\frac{g}{30}$
• C. $\frac{g}{5}$
• D. $\frac{g}{10}$

Answer 1

The correct option is C $\frac{g}{5}$

Velocity of transverse wave $v=\sqrt{\frac{T}{\mu }}$
$T$= tension in string
$\mu$= linear mass density
At the time when car is at rest
$60=\sqrt{\frac{Mg}{\mu }}$
In case when car is accelerating
$60.5=\sqrt{\frac{M(g^2+a^2{)}^{\frac{1}{2}}}{\mu }}$
$\Rightarrow \frac{60.5}{60}=\sqrt{\sqrt{\frac{g^2+a^2}{g^2}}}$ ${(1+\frac{0.5}{60})}^4=\frac{g^2+a^2}{g^2}=1+\frac{2}{60}$
(by using binomial expansion)

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

$a=g\sqrt{\frac{2}{60}}=\frac{g}{\sqrt{30}}=\frac{g}{5.47}\approx \frac{g}{5}$
Final answer: (a)