A given object takes n times more time to slide down a 45 o rough inclined plane as it takes to slide down a perfectly smooth 45 o incline. The coefficient of kinetic friction between the object and t

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A given object takes n times more time to slide down a 45 o rough inclined plane as it takes to slide down a perfectly smooth 45 o incline. The coefficient of kinetic friction between the object and the incline is:

TestHub · Accepted Answer

Time taken to slide along a smooth surface is $t_{1}$ .
$S = \frac{1}{2} g \sin 45^{o} t_{1}^{2}$
$t_{1} = \sqrt{\frac{2 \sqrt{2 s}}{g}}$
Time taken to slide along a rough surface is $t_{2}$ .
$S = \frac{1}{2} (g \sin 45^{o} - μ g \sin 45^{o}) t_{2}^{2}$
$t_{2} = \sqrt{\frac{2 \sqrt{2 s}}{g (1 - μ)}}$
$t_{2} = n t_{1}$
$\frac{2 \sqrt{2 s}}{g (1 - μ)} = n^{2} \times \frac{2 \sqrt{2 s}}{g} \Rightarrow 1 - μ = \frac{1}{n^{2}} \Rightarrow μ = 1 - \frac{1}{n^{2}}$

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