Question

A wheel of radius R is rolling in a straight line without slipping on a plane surface, the plane of the wheel is vertical. For the instant when the axis of the wheel is moving with a speed v relative to the surface, the instantaneous velocity of any point P on the rim of the wheel relative to the surface will be

Answer 1

Answer on Question #66180-Physics-Mechanics-Relativity

A wheel of radius $R$ is rolling in a straight line without slipping on a plane surface, the plane of the wheel is vertical. For the instant when the axis of the wheel is moving with a speed $v$ relative to the surface, the instantaneous velocity of any point $P$ on the rim of the wheel relative to the surface will be

Solution

Assume no slipping:

$$v _ {S} = 0 = \omega R - v \rightarrow \omega R = v$$

The instantaneous velocity of any point $P$ on the rim of the wheel relative to the surface will be

$$\vec {V} = \vec {v} + \overline {{\omega R}}$$ $$V = \sqrt {v ^ {2} + v ^ {2} + 2 v v \cos \theta} = v \sqrt {2 (1 + \cos \theta)}$$

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