A heavy ball of mass M is suspended from the ceiling of car by a light string of mass $\mathrm{m}(\mathrm{m} \ll \mathrm{M})$. When the car is at rest, the speed of transverse waves in the string is 60 $\mathrm{ms}^{-1}$. When the car has acceleration a , the wave-speed increases to $60.5 \mathrm{~ms}^{-1}$. The value of a, in terms of gravitational acceleration $g$ is closest to:

JEE Main · 2019 · 09 Jan 2019 · Shift 1 · Laws of Motion

A heavy ball of mass M is suspended from the ceiling of 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

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:

Answer: (B)

The transverse speed is

v = T / μ

. At rest

T = M g

; with acceleration the string settles along the effective gravity

g^{2} + a^{2}

, so

T^{'} = M g^{2} + a^{2}

. Then

\frac{v ^{'2}}{v ^{2}} = \frac{g ^{2} + a ^{2}}{g} = (\frac{60.5}{60})^{2} \approx 1.0167

, so

1 + (a / g)^{2} = 1.0167

, giving

(a / g)^{2} \approx 0.0338

and

a \approx 0.184 g

, closest to

\frac{g}{5}

JEE Main · 2019 · 09 Jan 2019 · Shift 1 · Single correct option