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  4. Three bodies a ring (R), a solid cylinder (C) and a solid sphere (S) having same mass and same radius roll down the inclined plane without slipping. They start from rest, if vR, vC and vS are velocities of respective bodies on reaching the bottom of the plane, then

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Q. Three bodies a ring $(R)$, a solid cylinder $(C)$ and a solid sphere $(S)$ having same mass and same radius roll down the inclined plane without slipping. They start from rest, if $v_R, v_C$ and $v_S$ are velocities of respective bodies on reaching the bottom of the plane, then

KCETKCET 2016System of Particles and Rotational Motion

Solution:

Velocity of ring $R=v_{R}$
Velocity of cylinder $C=v_{C}$
Velocity of sphere $S=v_{S}$
$m_{R}=m_{C}=m_{S}$
The velocity of body rolling without slipping down an inclined plane is given by the formula
$v-\sqrt{\frac{2 g h}{1+k^{2} / R^{2}}}$
The value of $k^{2} / R^{2}$ is minimum for sphere and maximum for ring.
$\therefore v_{S} > v_{C} > v_{R} $ or $ v_{R} < v_{C} < v_{S}$