AP® Physics C: Mechanics · Unit 5 · Topic 5.2
Connecting Linear and Angular Quantities
Every point on a rotating object shares the same angular velocity, but its linear speed depends on how far it sits from the axis: v = r omega. The farther out, the faster it moves through space.
A spinning object — a wheel, a disk, a merry-go-round — has one angular velocity omega shared by every point on it, because every point sweeps the same angle in the same time. But points at different distances from the axis trace different arc lengths doing that sweep. Arc length is s = r theta, so a point twice as far out covers twice the distance for the same angle. Divide by time and you get v = r omega: linear speed grows with radius even though angular speed stays fixed. The same logic applies one derivative up, giving tangential acceleration a = r alpha.
This is why a bigger bicycle wheel carries you farther per pedal stroke. Your legs set the angular speed of the wheel through the gears. A larger wheel means its rim — the part touching the road — sits farther from the axle, so that same omega produces a larger v, and the bike covers more ground for each rotation.
This conversion is the hinge between the rotational kinematics you just learned and the dynamics coming next. Torque, angular momentum, and rotational kinetic energy all get built from v = r omega and a = r alpha, translating linear concepts you already trust into their rotational counterparts. Keep the units straight: omega is in radians per second, not revolutions or degrees, or the conversion breaks.
Key terms
- angular velocity (omega)
- The rate an object sweeps angle, in radians per second, the same for every point on a rigid body.
- linear speed (v)
- The actual speed a specific point moves through space, equal to r omega.
- arc length (s)
- The distance traveled along a circular path, equal to r theta for angle theta in radians.
- radius (r)
- The distance from the rotation axis to the point in question; it scales linear quantities from angular ones.
Watch it explained
- How it worksHow does the arc length formula convert an angle swept into a distance traveled?
- Worked exampleFind the linear speed of a point 0.35 m from the axis of a wheel spinning at 12 rad/s.
- Worked exampleFind how fast a 0.66 m diameter tire spins when the car it carries moves at 27 m/s.
- Common mistakeWhy isn't every point on a spinning disk moving at the same speed?
Check yourself
End of topic.