Angular velocity and sign convention

Mechanical connections between rotating discs are everywhere: from bicycle gears to car transmissions, from windmills to mechanical clocks. Analysing them requires two quantities and a convention.

Every point on the rim of a disc rotating with angular velocity $\omega$ moves with tangential velocity

$v = \omega\,r$

where $r$ is the radius. This is the relation linking “how fast” the disc spins (a global property, $\omega$) to “how fast” a point on the rim moves (a local property, $v$, which grows with the radius).

To understand when two discs rotate “in the same direction” or “in opposite directions” it is essential to fix the sign of $\omega$:

Sign convention

$\omega > 0 \;\Leftrightarrow\; \text{anticlockwise rotation} \qquad \omega < 0 \;\Leftrightarrow\; \text{clockwise rotation}$

This convention is what distinguishes the three types of connection: two discs can have the same tangential velocity at the contact point ($|v_1| = |v_2|$) and yet spin in opposite directions (opposite signs of $\omega$) or in the same direction (matching signs). Keeping magnitude and sign of $\omega$ separate is the key to the whole chapter.

Links

Topics: Dinamica Concepts: Velocità angolare · Moto circolare uniforme Objects: Ingranaggi e dischi coassiali · Disco rotante

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