Question

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

• A. $\frac{y_1}{3}$
• B. $\frac{2y_1}{3}$
• C. $\frac{3y_1}{2}$
• D. $y_1$

Answer 1

Answer: (B)

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)
$mg{y}_1=\frac{3}{4}m{v}^2$
For part $BC$ , (there is no friction, hence no change in rotational kinetic energy)
$mg{y}_2=\frac{1}{2}m{v}^2$
Dividing both equations we get
$\therefore$ $y_2=\frac{2}{3}{y}_1$