Discs in contact: gears

Two discs in contact at their rims — like mechanical gears or two toothed wheels “biting” into each other — must have the same tangential velocity at the point of contact. If they did not, the teeth would slip against each other, or deform: the rigid contact requires the two rims to move together.

Principle — Discs in contact

Two discs in contact at their rims have tangential velocities equal in magnitude: $|v_1| = |v_2| \;\;\Rightarrow\;\; |\omega_1|\,r_1 = |\omega_2|\,r_2$ but $\omega_1$ and $\omega_2$ have opposite sign: if one turns clockwise, the other turns anticlockwise.

The opposite sign is the signature of gears: think of two toothed wheels biting into each other — when one pushes the teeth forward at the contact point, the other is forced to retreat. In terms of magnitudes, the relation $|\omega_1| r_1 = |\omega_2| r_2$ says that the smaller disc spins faster: a large gear coupled to a small one makes the small one spin with a larger $\omega$, but in the opposite direction.

Rule of thumb

Large gear + small gear: the small one spins faster (higher $\omega$) than the large one, but in the opposite direction.

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Topics: Dinamica Concepts: Velocità angolare Objects: Ingranaggi e dischi coassiali

Related exercises: Proof of centripetal acceleration · Centripetal acceleration of Earth’s rotation · Ranking points on a rotating disc