JEE Main · 2021 · 20 Jul 2021 · Shift 2 · Laws of Motion

A body of mass '

m

' is launched up on a rough inclined plane making an angle of

3 0^{\circ}

with the horizontal. The coefficient of friction between the body and plane is

\frac{x}{5}

if the time of ascent is half of the time of descent. The value of

x

is ____.

Answer: 3

Solution

Going up, gravity and friction both retard the body, so

a_{up} = g (sin θ + μ cos θ)

; coming down, friction acts up the plane and

a_{down} = g (sin θ - μ cos θ)

. The same distance is covered from rest each way, so

\frac{1}{2} a_{up} t_{up}^{2} = \frac{1}{2} a_{down} t_{down}^{2}

, and with

t_{up} = \frac{1}{2} t_{down}

this forces

a_{up} = 4 a_{down}

sin θ + μ cos θ = 4 (sin θ - μ cos θ) \Rightarrow 5 μ cos θ = 3 sin θ

μ = \frac{3}{5} tan 3 0^{\circ} = \frac{3}{5 3} = \frac{3}{5}

Comparing with

\frac{x}{5}

gives

x = 3

JEE Main · 2021 · 20 Jul 2021 · Shift 2 · Integer answer