A solid cylinder of mass M and radius R released from rest rolls down a rough inclined plane inclined at an angle of $$45$$∘ with the horizontal. The coefficient of friction between the cylinder and the inclined plane if the heat produced is maximum is $$1$$−n. Find n.

Question

A solid cylinder of mass M and radius R released from rest rolls down a rough inclined plane inclined at an angle of  $$45$$∘ with the horizontal. The coefficient of friction between the cylinder and the inclined plane if the heat produced is maximum is $$1$$−n. Find n.

Infinity Learn · Accepted Answer

Acceleration of cylinder  $$=g(sin45o$$−μ$$cos45o)=g2(1$$−μ$$)$$ Angular acceleration of cylinder  $$=$$τ$$frictionI=R$$μ$$mgcos45o(mR22)$$ $$=2$$μgRGain in $$KE$$ of cylinder travelling a distance $$L=12mv2+12I$$ω$$2=12m(2g2(1$$−μ$$)L)+12(mR22)22$$μ$$gR2$$μ$$L1-$$μR $$=mgL2$$[$$(1$$−μ$$)+2$$μ$$21$$−μ] Loss in $$PE$$ of cylinder $$=$$ $$mgh$$$$=mgL2Heat$$ generated $$=$$ Loss in $$PE$$ $$-$$ Gain in $$KE$$ $$=$$ $$mgL2-mgL2$$[$$(1$$−μ$$)+2$$μ$$21$$−μ]Since, this heat loss generated due to kinetic friction depends on μ,Differentiating this equation wrt μ we get,  μ$$=1$$−$$23$$

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