Source: High school physics (Chinese)
Problem
When a bicycle moves forward on horizontal ground with a speed $v$, what are the instantaneous speeds of the wheel's axle, the highest point on the wheel, and the lowest point on the wheel? Assume there is no slipping between the wheel and the ground.
- What is the instantaneous speed of the wheel's axle?
- What is the instantaneous speed of the highest point on the wheel?
- What is the instantaneous speed of the lowest point on the wheel?
The motion of a rolling wheel is the superposition of the translational motion of its center of mass (axle) and the rotational motion about its center. The speed of the bicycle, $v$, is the translational speed of the axle, $v_{cm}$.
For rolling without slipping, the tangential speed of any point on the rim relative to the center is $v_{tan} = \omega R = v_{cm} = v$.
The velocity of any point on the wheel is the vector sum of the translational velocity of the center, $\vec{v}_{cm}$, and the tangential velocity relative to the center, $\vec{v}_{tan}$.
Axle: The axle is the center of mass, its speed is the translational speed of the bicycle.
$$v_{axle} = v_{cm} = v$$Highest Point: The translational and tangential velocities are in the same direction.
$$v_{top} = v_{cm} + v_{tan} = v + v = 2v$$Lowest Point: The translational and tangential velocities are in opposite directions.
$$v_{bottom} = v_{cm} - v_{tan} = v - v = 0$$