Sankalp NEET Full Test-3 Question-4 Solution

Question: 4. There are two inclined surfaces of equal length L and same angle of inclination 45^{\circ} with the horizontal. One of them is rough and the other is perfectly smooth. A given body takes 2 times as much time to slide down on the rough surface than on the smooth surface. The coefficient of kinetic friction \mu_k between the object and the rough surface is close to

(1) 0.25

(2) 0.40

(3) 0.5

(4) 0.75

Answer: Option (4)

Explanation: For an incline of angle \theta, the acceleration on a smooth surface is

a_s = g\sin\theta

For a rough incline the acceleration is

a_r = g(\sin\theta - \mu_k\cos\theta)

The time to slide down length L with constant acceleration a is

t = \sqrt{\frac{2L}{a}}

Given that the rough surface takes twice the time:

t_r = 2t_s

Thus,

\sqrt{\frac{2L}{a_r}} = 2\sqrt{\frac{2L}{a_s}}

Squaring both sides:

\frac{1}{a_r} = 4\frac{1}{a_s}

So,

a_r = \frac{a_s}{4}

Substitute accelerations:

g(\sin\theta - \mu_k\cos\theta) = \frac{g\sin\theta}{4}

Cancel g:

\sin\theta - \mu_k\cos\theta = \frac{\sin\theta}{4}

Rearrange:

\sin\theta\left(1 - \frac{1}{4}\right) = \mu_k\cos\theta \frac{3}{4}\sin\theta = \mu_k\cos\theta

Thus,

\mu_k = \frac{3}{4}\tan\theta

For \theta = 45^\circ,

\tan45^\circ = 1

Therefore,

\mu_k = \frac{3}{4} = 0.75

The correct option is (4).