A smooth block is released at rest on a $$45$$° incline and then slides a distance d. The time taken to slide is n times as much to slide on rough incline than on a smooth incline. The coefficient of friction is

Question

Infinity Learn · Accepted Answer

See fig.Let t be the time taken by the block to cover a distance d.Then using the formula, $$s=ut+(1/2)g$$′$$t2$$, we get $$d=12g$$ $$sin$$⁡$$45$$∘$$t2$$   …(1)∵$$u=0$$  and  g′$$=gsin$$⁡$$45$$∘Let t' be the time taken for rough plane.Then $$d=12gsin$$⁡$$45$$∘−μkcos⁡$$45$$∘t′$$2$$ or $$d=12gsin$$⁡$$45$$∘−μkcos⁡$$45$$∘(nt)2$$    …$$(2)Equating$$ equations $$(1) and (2), we getsin⁡$$45$$∘$$=$$$$sin$$⁡$$45$$∘−μkcos⁡$$45$$∘$$n2or$$ $$12$$×$$1n2=12$$−μ$$k2$$  or  μ$$k=1$$−$$1n2$$.

A smooth block is released at rest on a 45° incline and then slides a distance d. The time taken to slide is n times as much to slide on rough incline than on a smooth incline. The coefficient of friction is

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