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MCQMediumJEE 2023Friction (Static, Kinetic, Rolling)

JEE Physics 2023 Question with Solution

The time taken by an object to slide down a 45∘45^\circ rough inclined plane is nn times as it takes to slide down a perfectly smooth 45∘45^\circ inclined plane. The coefficient of kinetic friction between the object and the inclined plane is:

  • A

    11−n2\sqrt{\frac{1}{1-n^2}}

  • B

    1−1n2\sqrt{1-\frac{1}{n^2}}

  • C

    1+1n21+\frac{1}{n^2}

  • D

    1−1n21-\frac{1}{n^2}

Answer

Correct answer:D

Step-by-step solution

Standard Method

Given: The incline angle is 45∘45^\circ. Let the acceleration on the smooth plane be a1a_1 and on the rough plane be a2a_2. The time on the rough plane is t2=nt1t_2 = n t_1.

Find: The coefficient of kinetic friction μ\mu.

For the smooth inclined plane:

a1=gsin⁡θ=g2a_1 = g \sin\theta = \frac{g}{\sqrt{2}}

For the rough inclined plane:

a2=gsin⁡θ−μgcos⁡θ=g2−μg2a_2 = g \sin\theta - \mu g \cos\theta = \frac{g}{\sqrt{2}} - \mu \frac{g}{\sqrt{2}}

Using the equation of motion for the same distance travelled:

a1t12=a2t22a_1 t_1^2 = a_2 t_2^2

Substituting t2=nt1t_2 = n t_1:

g2t12=(g2−μg2)(nt1)2\frac{g}{\sqrt{2}} t_1^2 = \left(\frac{g}{\sqrt{2}} - \mu \frac{g}{\sqrt{2}}\right)(n t_1)^2

Simplifying:

1=n2(1−μ)1 = n^2 (1 - \mu)

Therefore,

μ=1−1n2\mu = 1 - \frac{1}{n^2}

Therefore, the coefficient of kinetic friction is 1−1n21 - \frac{1}{n^2}. The correct option is D. The solution marks option C, but its own working gives D.

Common mistakes

  • Using the marked correct option without checking the derivation. The solution working clearly gives μ=1−1n2\mu = 1 - \frac{1}{n^2}, so the option mapping must follow the algebra, not the mislabeled key.

  • Writing the rough-plane acceleration as gsin⁡θ+μgcos⁡θg\sin\theta + \mu g\cos\theta. Friction opposes motion down the plane, so it must be subtracted: a2=gsin⁡θ−μgcos⁡θa_2 = g\sin\theta - \mu g\cos\theta.

  • Using a linear time-acceleration relation for fixed distance. For motion from rest over the same distance, the correct relation is through s=12at2s = \frac{1}{2}at^2, which leads to a1t12=a2t22a_1 t_1^2 = a_2 t_2^2.

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