JEE PYQ: Waves And Sound Question 38
Question 38 - 2019 (09 Jan Shift 1)
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 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:
(1) $\frac{g}{30}$
(2) $\frac{g}{5}$
(3) $\frac{g}{10}$
(4) $\frac{g}{20}$
Show Answer
Answer: (2)
Solution
Wave speed $V = \sqrt{\frac{T}{\mu}}$
When car is at rest $a = 0$:
$\therefore 60 = \sqrt{\frac{Mg}{\mu}}$
Similarly when the car is moving with acceleration $a$,
$60.5 = \sqrt{\frac{M(g^2 + a^2)^{1/2}}{\mu}}$
$\frac{60.5}{60} = \sqrt[4]{\frac{g^2 + a^2}{g^2}}$
$\left(1 + \frac{0.5}{60}\right)^4 = \frac{g^2 + a^2}{g^2} = 1 + \frac{a^2}{g^2}$
$\Rightarrow g^2 + a^2 = g^2 + g^2 \times \frac{2}{60}$
$a = g\sqrt{\frac{2}{60}} = \frac{g}{\sqrt{30}}$
which is closest to $g/5$.
BETA
