The upper half of an inclined plane with incl | vedclass

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(C) Let the total length of the inclined plane be $l$. The upper half has length $l/2$ and is smooth,while the lower half has length $l/2$ and is roug

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$2 	an \phi$

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(C) Let the total length of the inclined plane be $l$. The upper half has length $l/2$ and is smooth,while the lower half has length $l/2$ and is rough with coefficient of friction $\mu$.By the work-energy theorem,the total work done on the body is equal to the change in its kinetic energy.Since the body starts from rest and ends at rest,the change in kinetic energy is $0$.Work done by gravity = Work done against friction.The vertical height of the entire plane is $h = l \sin \phi$.The work done by gravity is $W_g = mgh = mgl \sin \phi$.The work done against friction on the lower half is $W_f = f_k 	imes (l/2) = \mu N 	imes (l/2) = \mu (mg \cos \phi) (l/2)$.Equating the two: $mgl \sin \phi = \mu mg \cos \phi (l/2)$.Simplifying the equation: $\sin \phi = \mu \frac{\cos \phi}{2}$.Therefore,$\mu = 2 \frac{\sin \phi}{\cos \phi} = 2 	an \phi$.

The upper half of an inclined plane with inclination $\phi$ is perfectly smooth,while the lower half is rough. $A$ body starting from rest at the top will again come to rest at the bottom,if the coefficient of friction for the lower half is given by-

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