Newtonian mechanics

Uniform circular motion and centripetal force

In uniform circular motion, speed is constant but velocity changes direction; centripetal acceleration v²/r requires a net inward force.

An object moving uniformly around a circle does not increase its speed, but continuously changes direction. Since velocity changes, it accelerates even at constant speed.

Acceleration from changing direction

Over a very short interval, two velocity vectors have nearly equal magnitudes but different directions. Their difference points inward, yielding $a_c=v^2/r$. The period is $T=2\pi/\omega$ and frequency $f=1/T$.

Quick check

To find the centripetal force, draw a free-body diagram and resolve forces along the radial direction toward the center; do not add a separate “centripetal force.” A car of mass 800 kg moving at 15 m/s around a 45 m bend needs an inward force $F=mv^2/r=4.0 imes10^3$ N. On a level road, tire–road friction supplies it; without enough friction the car skids away from the circular path.

At every point, velocity is tangent to the circle while acceleration points inward, so the two vectors are perpendicular in uniform circular motion. One revolution covers distance $2pi r$, giving $v=2pi r/T$ from the period $T$. Doubling speed at fixed radius quadruples the required acceleration because it depends on $v^2$.

Angular speed $omega$ is swept angle per time, measured in rad/s. At fixed radius, constant angular speed also means constant linear speed. Convert revolutions per minute to rad/s by multiplying by $2pi/60$.