Velocity In Rolling Motion

Question

A wheel of radius 0.4 m rolls without slipping along a level road while the speed of its centre is 5 m/s. What is the instantaneous speed of the topmost point on the rim of the wheel?

RankGuru · Accepted Answer

In rolling without slipping, the contact point with the ground is instantaneously at rest, and every other point's velocity is the vector sum of the centre's translation $v$ and the rotation $\omega r$ about the centre. The no-slip condition fixes $\omega = v/R$, so the rim speed relative to the centre equals $v$ itself. At the topmost point the translational velocity $v$ and the rotational velocity $\omega R = v$ point in the same forward direction and therefore add: $v_{top} = v + v = 2v = 2 	imes 5 = 10	ext{ m/s}$. The value 5 m/s gives only the centre's speed and ignores the rotational contribution. The value 0 m/s describes the bottom contact point, not the top. The value 7.5 m/s uses an incorrect intermediate factor of 1.5. This is the NCERT velocity analysis of a rolling wheel. As a check, the top must be the fastest point of the wheel, exceeding the centre's 5 m/s, and 10 m/s correctly captures that maximum. More generally, every point on the rim traces a cycloidal path as the wheel advances, with its speed ranging from zero at the ground contact to twice the hub speed at the very top, while a point level with the axle moves at $\sqrt{2}\,v$.

Velocity In Rolling Motion

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