Question

The time taken by an object to slide down 45° rough inclined plane is n times as it takes to slide down a perfectly smooth 45° incline plane. The coefficient of kinetic friction between the object and the incline plane is:

• A. \(\sqrt{1−\frac1{𝑛_2} }\)
• B. \(1+ \frac{1}{𝑛_2} \)
• C. \(1- \frac{1}{𝑛_2} \)
• D. \(\sqrt{ \frac{1}{1-𝑛^2}}\)

Answer 1

The Correct Option is C

For the smooth inclined plane: \[ a_1 = g \sin\theta = \frac{g}{\sqrt{2}} \] For the rough inclined plane: \[ a_2 = g \sin\theta - \mu g \cos\theta = \frac{g}{\sqrt{2}} - \mu \frac{g}{\sqrt{2}} \] Using \(t_2 = n t_1\) and the equation of motion: \[ a_1 t_1^2 = a_2 t_2^2 \] \[ \frac{g}{\sqrt{2}} t_1^2 = \left(\frac{g}{\sqrt{2}} - \mu \frac{g}{\sqrt{2}}\right)(n t_1)^2 \] Simplifying: \[ 1 = n^2 (1 - \mu) \quad \Rightarrow \quad \mu = 1 - \frac{1}{n^2} \]