Centripetal Force Worked Examples Cheat Sheet

Key Facts

• Centripetal acceleration is directed toward the center of the circle and has magnitude $a_c = \frac{v^2}{r}$.
• The net inward force required for circular motion is $F_c = ma_c = \frac{mv^2}{r}$.
• If period is given, circular speed is $v = \frac{2\pi r}{T}$, where $T$ is the time for one complete revolution.
• Centripetal force can also be written as $F_c = \frac{4\pi^2mr}{T^2}$ when mass, radius, and period are known.
• For uniform circular motion, speed $v$ is constant but velocity changes because the direction changes continuously.
• The centripetal force is always the net force toward the center, so it may be supplied by tension, friction, gravity, or a normal force.
• Increasing speed has a strong effect because centripetal force depends on $v^2$, so doubling $v$ makes $F_c$ four times larger.
• Before calculating, convert all quantities to SI units such as kilograms, meters, seconds, meters per second, and newtons.

Vocabulary

Centripetal force

The net inward force that keeps an object moving in a circular path.

Centripetal acceleration

The inward acceleration of an object in circular motion, given by $a_c = \frac{v^2}{r}$.

Radius

The distance $r$ from the center of the circle to the moving object.

Period

The time $T$ required for one complete revolution around a circle.

Uniform circular motion

Motion in a circle at constant speed where the direction of velocity changes continuously.

Tangential velocity

The velocity directed along the tangent to the circular path, with magnitude $v = \frac{2\pi r}{T}$.