How to Draw a Free-Body Diagram for Circular Motion

A free-body diagram for circular motion is made the same way as any other: isolate the object and draw every real force acting on it. The key extra step is to choose a radial axis pointing toward the center of the circular path. The forces’ inward components must provide the required inward acceleration.

“Centripetal force” describes the net inward force, not an additional force to add to the diagram. Depending on the situation, that net force may come from tension, friction, gravity, a normal force, or a combination of forces.

The radial force equation

For an object moving at speed $v$ along a circle of radius $r$, its radial acceleration points toward the center and has magnitude

$$a_r = \frac{v^2}{r}.$$

Newton’s second law in the radial direction is therefore

$$\sum F_r = m\frac{v^2}{r},$$

where inward is taken as positive. In this equation, each force contributes its component along the inward direction. A force pointing outward has a negative radial component. If the speed is changing, there is also a tangential acceleration, so apply Newton’s second law along the tangent as well.

How to draw the diagram

1. Choose the object. Represent it as a dot or simple shape, and leave out everything else.
2. Draw the real forces. Include gravity, contact forces, tension, friction, or applied forces as appropriate. Do not draw a separate “centripetal force” arrow.
3. Mark the center and radial direction. Indicate the inward radial axis with an axis arrow, not a force arrow. Choose a tangential axis too if the speed changes or you need to resolve forces.
4. Resolve and apply Newton’s second law. Add the signed inward components and set their sum equal to $mv^2/r$.

The center is not necessarily in the direction of any one force. For example, an object can have gravity downward and tension diagonally upward, while their combined force points inward.

Worked example: a car turning on a level road

A $1000\,\text{kg}$ car travels at $10\,\text{m/s}$ around a level curve of radius $50\,\text{m}$. Assume the only horizontal force is the static friction between the tires and the road.

The free-body diagram has three forces:

• Weight $mg$ downward.
• Normal force $N$ upward.
• Static friction $f_s$ horizontally toward the center of the turn.

There is no vertical acceleration, so $N-mg=0$. Horizontally, friction supplies the inward force:

$$\begin{aligned} f_s &= m\frac{v^2}{r}\\ &= (1000\,\text{kg})\frac{(10\,\text{m/s})^2}{50\,\text{m}}\\ &= 2000\,\text{N}. \end{aligned}$$

Thus the required static friction is $2000\,\text{N}$ toward the center. The road can provide this force only if the available tire-road friction is large enough; if it is not, the car cannot follow that curve at that speed without skidding.

Example: an object on a string in a vertical circle

The radial direction changes as the object moves around a vertical circle. At the top, inward points downward. Both tension and weight point inward, so

$$T+mg=m\frac{v^2}{r}.$$

At the bottom, inward points upward. Tension points inward, but weight points outward, giving

$$T-mg=m\frac{v^2}{r}.$$

The diagram still contains only tension and weight at both positions. What changes is the direction chosen as inward and, consequently, the signs in the radial equation.

Common mistakes

• Adding “centripetal force” as another arrow. Identify the physical forces first. Their net radial component is the centripetal force.
• Assuming there is always a force pointing exactly toward the center. The inward net force can result from components of several forces.
• Using $mv^2/r$ as a force on its own. It is the required net radial force, not necessarily the magnitude of any one force.
• Forgetting signs. Decide which direction is positive before writing the radial equation. In a vertical circle, gravity can point inward at the top and outward at the bottom.
• Confusing radial and tangential motion. The radial equation describes the change in direction of velocity. A net tangential force changes the speed.

For any circular-motion problem, draw the physical forces first, then use the center of the path to determine which components contribute to the inward radial equation.