Belts, Pulleys, and Rolling Contact

A belt wraps around two pulleys and drags their rims along with it. The belt itself only moves at one speed, since it's a single loop with no stretch or slipping. That means every point on the belt, and so every point on the rim of each pulley touching it, moves at that same linear speed v. Since v = omega r for each pulley, a small radius needs a large omega to produce the same v. That's why a tiny gear spins fast when a big gear drives it slowly: both rims move at the same speed, but the small one has to turn more times per second to cover that speed.

The key move is always the same: write v = omega1 r1 = omega2 r2, then solve for whatever is missing. For a 4 cm pulley driven by a belt from a 12 cm pulley spinning at 30 rad/s, the rim speed is v = (30)(0.12) = 3.6 m/s. The small pulley must match that rim speed, so omega2 = v / r2 = 3.6 / 0.04 = 90 rad/s. The radius ratio does the work directly: omega2 = omega1 (r1/r2) = 30(12/4) = 90 rad/s.

This same v = omega r link is what connects rolling without slipping back in Unit 2 and shows up again when a rope unwinds from a spool. If the spool's radius shrinks as rope comes off, the same angular speed now produces a smaller rope speed, since v = omega r falls as r falls. Rotational dynamics keeps reusing this one equation to translate between the spinning world and the straight-line world, the same way it reused force and mass from Unit 1 as torque and rotational inertia.