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 $60m{s}^{-1}$. When the car has acceleration $a$, the wave-speed increases to $60.5m{s}^{-1}$. The value of $a$, in terms of gravitational acceleration $g$, is closest to :

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

Answer 1

Answer: (A)

$60=\sqrt{\frac{Mg}{\mu }}$
$60.5=\sqrt{\frac{M{\left(g^2+a^2\right)}^{1/2}}{\mu }}\Rightarrow \frac{60.5}{60}=\sqrt{\sqrt{\frac{g^2+a^2}{g^2}}}$
${\left(1+\frac{0.5}{60}\right)}^4=\frac{g^2+a^2}{g^2}=1+\frac{2}{60}$
$\Rightarrow g^2+a^2=g^2+g^2\times \frac{2}{60}$
$a=g\sqrt{\frac{2}{60}}=\frac{g}{\sqrt{30}}=\frac{g}{5.47}$
$\simeq \frac{g}{5}$