First slide

Rolling motion

Question

The solid cylinder shown in figure is rolling without slipping between two moving planks P₁ and P₂ moving with velocities 2V and V respectively. If mass of the cylinder is m, then kinetic energy of the cylinder relative to ground frame is

Difficult

A

$\frac{1}{8} {mV}^{2}$
B

$\frac{11}{16} {mV}^{2}$
C

$\frac{7}{16} {mV}^{2}$
D

$\frac{5}{8} {mV}^{2}$

Solution

Angular velocity of the cylinder $ω = \frac{2 V + V}{2 R} = \frac{3 V}{2 R}$
$∴$ Velocity of centre C is given by ${\vec{V}}_{C} = {\vec{V}}_{B} + \frac{{\vec{V}}_{C}}{B}$
$∴$ Kinetic energy relative to ground frame
$= \frac{1}{2} {mV}_{C}^{2} + \frac{1}{2} I_{C} ω^{2} = \frac{1}{2} . m {(\frac{V}{2})}^{2} + \frac{1}{2} (\frac{1}{2} {mR}^{2}) . {(\frac{3 V}{2 R})}^{2}$
$= \frac{11}{16} {mV}^{2}$

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