Uniform Circular Motion

Constant Speed Still Changes Direction

Uniform circular motion is motion along a circular path with constant speed. The speed stays the same, while the direction of motion keeps changing because the object follows a curved path.

Picture a small car moving around a round track. Every second, the car can cover the same length of track. But the car's velocity on the right side of the track points in a different direction from its velocity at the top of the track.

In uniform circular motion, speed is constant, but velocity still changes because its direction changes.

Speed describes how fast the object moves along the path. Velocity describes both speed and direction.

Car on a Circular Track

Change the period to see the car complete each lap at a different speed.

Period: $6\text{ s}$
Radius: $4\text{ m}$
Speed: $4.19\text{ m/s}$
Centripetal acceleration: $4.39\text{ m/s}^2$

Angular Position and Displacement

In straight-line motion, a coordinate gives the position relative to zero. In circular motion, an angle gives the position relative to a reference line. The change in that angle is called angular displacement.

If an object moves from a reference line and sweeps a signed angle $\Delta\theta$ in time $\Delta t$ , its average angular velocity is:

\omega_{\text{avg}}=\frac{\Delta\theta}{\Delta t}

The symbol $\omega$ is called omega. In uniform circular motion, angular velocity is constant, so its value equals the average over every interval. Its unit is radians per second, written $\text{rad/s}$ . Radians are used because one full turn equals $2\pi$ radians.

Period and Frequency Describe One Revolution

The period, written $T$ , is the time needed for one revolution. The frequency, written $f$ , is the number of revolutions completed each second.

They are reciprocals:

f=\frac{1}{T}

When the period is smaller, the object completes a revolution faster. As a result, its frequency and angular speed $\lvert\omega\rvert$ are larger. The sign of $\omega$ separately records the rotation direction.

Velocity and Acceleration Directions on a Circle

At every point on a circular path, velocity points tangent to the circle. These two directions are perpendicular, so an object can have constant speed and still have acceleration.

Imagine a small object on the right side of the circle moving counterclockwise. Its velocity points upward along the path, while its acceleration points left toward the center. This direction pair is what makes circular motion different from ordinary straight-line motion.

Tangential Speed Connects to Angular Speed

Tangential speed is the magnitude of the object's velocity along the circular path. If the radius is $r$ , one full circumference is $2\pi r$ . Because one revolution takes time $T$ , the tangential speed is:

v=\frac{2\pi r}{T}

The formula $v=\lvert\omega\rvert r$ means two objects with the same angular speed do not always have the same tangential speed. The object farther from the center travels a longer path, so its tangential speed is larger.

Centripetal Acceleration Always Points Inward

Even though speed is constant, the velocity direction keeps turning. Changing the direction of velocity requires acceleration. In uniform circular motion, this acceleration is called centripetal acceleration.

a_c=\frac{v^2}{r}

Centripetal acceleration always points toward the center of the circle. So do not think of this acceleration as pointing along the object's motion. Velocity is tangent to the path, while centripetal acceleration points inward.

One Full Turn of a Small Car

Suppose a small car moves on a track with radius $4\text{ m}$ and completes one revolution in $6\text{ s}$ .

Its angular velocity is:

\lvert\omega\rvert=\frac{2\pi}{T}=\frac{2\pi}{6}=\frac{\pi}{3}\text{ rad/s}

Its tangential speed is:

\begin{aligned} v &=\lvert\omega\rvert r \\ &=\frac{\pi}{3}(4) \\ &=\frac{4\pi}{3}\text{ m/s} \\ &\approx4.19\text{ m/s} \end{aligned}

Its centripetal acceleration is:

a_c=\frac{v^2}{r}=\frac{4\pi^2}{9}\text{ m/s}^2\approx4.39\text{ m/s}^2

So the car moves with constant speed of about $4.19\text{ m/s}$ , but it still has acceleration of about $4.39\text{ m/s}^2$ because the direction of its velocity keeps changing.