AI Tutor
When you shift a bicycle into high gear, your pedaling barely speeds up, yet the bike shoots down the road twice as fast.
This mechanical magic happens because interconnected rotating parts can trade rotational speed for linear speed at will.
How do we mathematically track this motion from one spinning wheel to the next?

Linear Speed vs. Angular Speed

Every spinning wheel has two distinct velocities: angular velocity (Ο‰\omega), which measures how fast an angle opens up in radians per second (extrad/s ext{rad/s}), and tangential speed (vv), which measures how fast a point on the rim travels along its circular path in meters per second (extm/s ext{m/s}).
Because arc length along a circle is s=rhetas = r heta, dividing both sides by time tt proves that tangential rim speed is directly proportional to radius: v=rωv = r\omega
If two separate wheels are linked by a belt, how does the motion of the first wheel govern the second?

Belt and Pulley Systems

Assuming a flexible belt does not stretch or slip, every millimeter of the belt moves at the exact same speed throughout the entire loop.
Therefore, the tangential speeds at the outer rims of both connected pulleys must be identical: v1=v2β€…β€ŠβŸΉβ€…β€Šr1Ο‰1=r2Ο‰2v_1 = v_2 \implies r_1\omega_1 = r_2\omega_2
Notice the inverse relationship: a smaller pulley must spin with a larger angular velocity Ο‰\omega to keep up with the belt's speed.