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  4. A uniform cylinder rolls down from rest, on a track whose vertical cross-section is a parabola given by the equation y=kx2 . 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 <img class=img-fluid question-image alt=Question src=https://cdn.tardigrade.in/q/nta/p-l9uu1vx8tb84.png />

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Q. A uniform cylinder rolls down from rest, on a track whose vertical cross-section is a parabola given by the equation $y=kx^{2}$ . 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

Question

NTA AbhyasNTA Abhyas 2020System of Particles and Rotational Motion

Solution:

Solution
Applying work-energy theorem
Work done by gravity $=$ Change in Kinetic Energy
For part $AB$ , ( 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
$\therefore $ $y_{2}=\frac{2}{3}y_{1}$