Laws of Motion

JEE Main level, single correct, medium

177 of 210

Question 177 of 210

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

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Answer: (B) 1−1n21-\dfrac{1}{{n}^{2}}

Time taken to slide along a smooth surface is t1{t}_{1}.
S=12gsin⁡⁡45ot12S=\dfrac{1}{2}g\sin ⁡{45}^{o}{t}_{1}^{2}
t1=22sg{t}_{1}=\sqrt{\dfrac{2\sqrt{2s}}{g}}
Time taken to slide along a rough surface is t2{t}_{2}.
S=12(gsin⁡⁡45o−μgsin⁡⁡45o)t22S=\dfrac{1}{2}\left(g\sin ⁡{45}^{o}-\mu g\sin ⁡{45}^{o}\right){t}_{2}^{2}
t2=22sg(1−μ){t}_{2}=\sqrt{\dfrac{2\sqrt{2s}}{g\left(1-\mu \right)}}
t2=nt1{t}_{2}=n{t}_{1}
22sg(1−μ)=n2×22sg⇒1−μ=1n2⇒μ=1−1n2\dfrac{2\sqrt{2s}}{g\left(1-\mu \right)}={n}^{2}\times \dfrac{2\sqrt{2s}}{g}\Rightarrow 1-\mu =\dfrac{1}{{n}^{2}}\Rightarrow \mu =1-\dfrac{1}{{n}^{2}}

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