Describe uniform circular motion, calculate centripetal acceleration with $a_c = v^2/r$ and centripetal force with $F_c = mv^2/r$, and identify the real force that supplies the centripetal force in a given situation.

What this topic is asking

Uniform circular motion is motion in a circle at constant speed, and the Regents tests one central, counterintuitive idea: such motion is accelerated, because the direction of the velocity is always changing. The Physical Setting/Physics course asks you to calculate the centripetal acceleration and centripetal force directed toward the center, and to identify which real force (tension, gravity, friction, the normal force) supplies that centripetal force in a given situation.

Why circular motion is accelerated

This is the conceptual heart of the topic. Speed (a scalar) does not change, so kinetic energy does not change, but velocity (a vector) does, so there is acceleration. Because the acceleration is always perpendicular to the motion, it changes only the direction of the velocity, not its size, which is exactly why the speed stays constant.

Centripetal acceleration and force

The centripetal force is just Newton's second law applied to circular motion: $F_c = ma_c = m\dfrac{v^2}{r}$. Note the strong dependence on speed (squared) and the inverse dependence on radius: a tighter turn (smaller $r$) or a faster speed needs a much larger inward force. This is why fast, tight turns are hard to make.

What supplies the centripetal force

A frequent Regents error is to invent an outward "centrifugal force". There is no such force on the object; the outward feeling (being pressed against a car door on a turn) is the body's inertia resisting the inward acceleration. The only force on the object is inward.

Reference Tables note

The Reference Tables print $a_c = \dfrac{v^2}{r}$, $F_c = \dfrac{mv^2}{r}$ and $v = \dfrac{2\pi r}{T}$ in the Mechanics section. You supply the recognition that circular motion is accelerated, the identification of the real force providing the centripetal force, and the understanding that no work is done by the centripetal force (it is perpendicular to the velocity), so the speed and kinetic energy are constant, as discussed in work and power.

Try this

Q1. State why an object moving at constant speed in a circle is accelerating. [2 points]

• Cue. Its velocity direction is constantly changing; a changing velocity (a vector) means acceleration, directed toward the center.

Q2. A $2.0$ kg object moves in a circle of radius $4.0$ m at $6.0$ m/s. Calculate the centripetal force. [2 points]

• Cue. $F_c = \dfrac{mv^2}{r} = \dfrac{(2.0)(6.0)^2}{4.0} = \dfrac{(2.0)(36)}{4.0} = 18$ N toward the center.