A uniform cylinder rolls down from rest, on a track whose vertical cross-section is a parabola given by the equation $y = k x^{2}$ . If the surface is rough from $A$ to $B$ due to which the cylinder doesn't slip but it is frictionless from $B$ to $C$ , then the height of ascent of cylinder towards $C$ is

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TestHub · Accepted Answer

Applying work-energy theorem

Work done by gravity $=$ Change in Kinetic Energy

For part $A B$ , ( friction will provide torque to generate rotational kinetic energy but work done by friction will be zero because the velocity of the point of action of friction is zero)

${mgy}_{1} = \frac{3}{4} {mv}^{2}$

For part $BC$ , (there is no friction, hence no change in rotational kinetic energy)

${mgy}_{2} = \frac{1}{2} {mv}^{2}$

Dividing both equations we get
$∴$ $y_{2} = \frac{2}{3} y_{1}$

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