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

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

Answer 1

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

$\text{Wave speed},V=\sqrt{\frac{T}{\mu }}$

When car is at rest $a=0\Rightarrow T=mg$

$\therefore 60=\sqrt{\frac{Mg}{\mu }}--\left(1\right)$

Similarly, when the car is moving with acceleration $a$, a horizontal pseudo force acts on the mass in addition to vertical $mg$,

$\Rightarrow T^{\prime }=\sqrt{(Mg{)}^2+(Ma{)}^2}$

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

Dividing $\left(2\right)$ by $\left(1\right)$

$\frac{60.5}{60}=\sqrt{\sqrt{\frac{g^2+a^2}{g^2}}}$

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

Using Binomial expansion,

$1+\frac{2}{60}=\frac{g^2+a^2}{g^2}$

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

$\Rightarrow a=g\sqrt{\frac{2}{60}}=\frac{g}{\sqrt{30}}$

Which is closest to $\frac{g}{5}$

Hence, option $\left(B\right)$ is correct.