JEE Main · 2023 · 29 Jan 2023 · Shift 2 · Laws of Motion

The time taken by an object to slide down

4 5^{\circ}

rough inclined plane is

n

times as it takes to slide down a perfectly smooth

4 5^{\circ}

incline plane. The coefficient of kinetic friction between the object and the incline plane is

Answer: (D)

Solution

On the smooth incline the acceleration is

a_{1} = g sin 4 5^{\circ}

, while on the rough incline it is

a_{2} = g sin 4 5^{\circ} - μ_{k} g cos 4 5^{\circ} = g sin 4 5^{\circ} (1 - μ_{k})

, since

sin 4 5^{\circ} = cos 4 5^{\circ}

. For the same slope length,

s = \frac{1}{2} a t^{2}

gives

a_{1} t_{1}^{2} = a_{2} t_{2}^{2}

, and with

t_{2} = n t_{1}

this becomes

a_{1} = a_{2} n^{2}

. Then

g sin 4 5^{\circ} = g sin 4 5^{\circ} (1 - μ_{k}) n^{2}

leads to

(1 - μ_{k}) n^{2} = 1

, so the coefficient of kinetic friction is

μ_{k} = 1 - \frac{1}{n ^{2}}

JEE Main · 2023 · 29 Jan 2023 · Shift 2 · Single correct option