Summary table of the three cases

The three types of connection between rotating discs are distinguished by just two questions: is the tangential velocity shared? and do the signs of $\omega$ match?

The three types of connection between rotating discs. Gears: opposite directions, $|v_1| = |v_2|$. Coaxial: same $\omega$, $|v_1| \neq |v_2|$. Pulley: same directions, $|v_1| = |v_2|$.

Type	Angular velocity $\omega$	Tangential velocity $v = \omega r$
Gears (in contact)	$\abs{\omega_1}r_1 = \abs{\omega_2}r_2$, opposite sign	$\abs{v_1} = \abs{v_2}$
Coaxial (rigidly coupled)	$\omega_1 = \omega_2$ (same sign and magnitude)	$\abs{v_1} \neq \abs{v_2}$ (depends on $r$)
Pulley (belt)	$\abs{\omega_1}r_1 = \abs{\omega_2}r_2$, same sign	$\abs{v_1} = \abs{v_2}$

How to remember them

• Rim contact (gears): $v$ equal, $\omega$ opposite.
• Rigidly coupled (coaxial): $\omega$ equal, $v$ different.
• Belt: $v$ equal, $\omega$ same sign.

The same relation holds for angular accelerations

All these rules also hold for the angular accelerations $\alpha_i$: if $\omega_1 r_1 = \omega_2 r_2$, differentiating with respect to time gives $\alpha_1 r_1 = \alpha_2 r_2$. So in accelerated-motion problems the rules “carry over” automatically to the $\alpha$‘s.

The complete example combining all three types of connection in a single system is worked out in Four discs chained together.

Links

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

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