Problem — Light wheels on racing bicycles

Problem

Why do racing bicycles have very light wheels with mass concentrated towards the hub? Explain in terms of moment of inertia and rotational kinetic energy, comparing with a wheel whose mass is concentrated at the outer rim.

Solution

The moment of inertia $I = \sum m_i r_i^2$ weights the mass by the square of the distance from the axis. Concentrating the mass near the hub (small $r$) makes $I$ very small; concentrating it at the rim (large $r$) makes it large — for the same total mass, a ring has $I = mR^2$, while a solid disc has only $\tfrac{1}{2}mR^2$.

A small moment of inertia gives two concrete advantages in a race:

• at the same riding speed, the rotational kinetic energy $E_\text{cin,rot} = \tfrac{1}{2}I\omega^2$ stored in the wheels is smaller, so when accelerating the cyclist spends less energy “spinning up” the wheels;
• less torque is needed to change the angular velocity (accelerating or braking), because $\tau = I\alpha$.

$\ev{\text{mass near the axis} \Rightarrow I \text{ small} \Rightarrow \text{less } E_\text{cin,rot} \text{ and less torque to accelerate}}$

Links

Topics: Dinamica rotazionale Concepts: Momento d’inerzia · Energia cinetica rotazionale Objects: Disco rotante