Problem — Fermi estimate: tyre on the motorway

Problem

Estimate the angular velocity of a tyre of radius $R = 30\;\mathrm{cm}$ on a car travelling at $120\;\mathrm{km/h}$.

Solution

The tyre rolls without slipping, so the constraint $v = \omega R$ holds. Convert the speed: $120\;\mathrm{km/h} = \dfrac{120\,000}{3600} \approx 33{,}3\;\mathrm{m/s}$. $\omega = \frac{v}{R} = \frac{33{,}3}{0{,}30} \approx \ev{111\;\mathrm{rad/s}}$ In revolutions per minute: $\omega \cdot \dfrac{60}{2\pi} \approx 1060\;\mathrm{rpm}$. At these speeds even a small mass imbalance in the rim is felt as vibration at the steering wheel: this is why wheels must be balanced.

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Topics: Dinamica rotazionale Concepts: Velocità angolare · Rotolamento Skills: Stime di Fermi