Topic 2.9 Circular Motion: analyze uniform circular motion using centripetal acceleration and the net inward (centripetal) force that produces it.

What this topic is asking

The College Board (Topic 2.9) wants you to analyze uniform circular motion: motion in a circle at constant speed. You must explain why such an object is accelerating even though its speed is constant, calculate its centripetal acceleration, and identify the net inward force (the centripetal force) that produces it. The key conceptual hurdle is accepting that "centripetal force" is not a new kind of force but the name for whatever real force points toward the center.

Why circular motion is accelerated motion

This is the conceptual heart of the topic. Velocity is a vector; turning it, even without changing its length, is a change in velocity, and any change in velocity is an acceleration. So an object circling at steady speed is accelerating at every instant.

Centripetal acceleration

The direction is always toward the center, perpendicular to the velocity (which is tangent to the circle). This perpendicular relationship is exactly why the speed stays constant: a force perpendicular to the motion changes direction but does no work to change speed.

The centripetal force

By Newton's second law, an inward acceleration requires an inward net force:

This centripetal force is the net force, directed toward the center. The vital point is that it is not a new force; it is supplied by whatever real force happens to point inward in a given situation:

• For a ball on a string, the tension provides it.
• For a satellite or planet, gravity provides it.
• For a car on a flat curve, friction provides it.
• For a car on a banked track or a rider in a loop, a component of the normal force provides it.

To solve a circular-motion problem, draw the free-body diagram, find the net force directed toward the center, and set it equal to $\dfrac{mv^2}{r}$.

Why there is no outward force

A persistent misconception is that an object in circular motion feels an outward "centrifugal" force. There is no such force acting on the object. What you feel when a car turns is your own inertia: your body tends to continue in a straight line (Newton's first law), and the door or seatbelt pushes you inward to make you follow the curve. The only real horizontal force on you is that inward push, which is the centripetal force. If the inward force is suddenly removed, for example a string snaps, the object does not fly outward; it flies off tangentially, in a straight line, exactly as the first law predicts. Period and speed are also connected: for one full circle the object travels a distance $2\pi r$ in one period $T$, so $v = \dfrac{2\pi r}{T}$, which lets you swap between speed and period in problems about orbits, turntables, or spinning rides. Keeping the force inward, recognizing it as a real named force, and remembering the tangential escape route, are the three ideas that make circular-motion problems reliable.

Try this

Q1. A $0.50$ kg object moves in a circle of radius $2.0$ m at $4.0$ m/s. Calculate the centripetal force. [2 points]

• Cue. $F_c = \dfrac{mv^2}{r} = \dfrac{(0.50)(4.0)^2}{2.0} = 4.0$ N toward the center.

Q2. State the direction in which a ball on a string flies if the string suddenly breaks. [1 point]

• Cue. Tangentially (in a straight line along its velocity at that instant), not outward.