5.8 Rotational and Translational Motion Relationships

Linking Linear and Angular Quantities

For an object rolling or rotating without slipping, linear quantities relate to angular ones through the radius r: v = rω, a = rα, and s = rθ. These equations connect a point's motion around an axis to its straight-line motion, letting you convert between rotational and translational kinematics whenever there's a fixed rotation radius.

Rolling Without Slipping

Rolling without slipping means the contact point between an object and a surface is momentarily at rest — no relative sliding occurs. This condition enforces v_cm = rω for the center of mass velocity and angular velocity. It's essential for problems involving wheels, balls, or cylinders rolling down inclines, since static friction (not kinetic) provides the torque without doing negative work via slipping.

Combining Translational and Rotational Kinetic Energy

A rolling object has both translational KE (½mv²) and rotational KE (½Iω²). Total kinetic energy is the sum of both. Using the rolling-without-slipping condition (v = rω), you can express total KE in terms of one variable, which is key for energy conservation problems like objects rolling down ramps.

Key terms

v = rω

Relates linear speed (v) to angular speed (ω) via radius (r) for rotational motion.

a = rα

Relates linear (tangential) acceleration to angular acceleration via radius r.

Rolling without slipping

Condition where the contact point between object and surface has zero relative velocity; v_cm = rω.

Role of friction in rolling

Static friction (not kinetic) provides the torque needed for rolling without slipping, doing no work on the object.

Total KE of a rolling object

KE_total = ½mv² (translational) + ½Iω² (rotational).

Try an AP-style question

A point on a wheel's rim is 0.5 m from the axis, and the wheel spins at 10 rad/s. What is its speed?

1. 20 m/s
2. 10 m/s
3. 0.05 m/s
4. 5 m/s