2.1.5.4. Uniform Circular Motion

Table of Contents

Constant speed on a circle

Uniform circular motion is motion along a circular path with constant speed. The direction of motion changes continuously, even though the magnitude of the velocity stays the same. This makes uniform circular motion different from motion in a straight line at constant speed.

If an object moves around a circle of radius $r$ and completes each revolution in the same amount of time, its motion is uniform circular motion. Common examples are a point on a rotating wheel, a satellite in a circular orbit, or a stone tied to a string and swung in a circle.

What stays constant and what changes

In uniform circular motion, several quantities remain constant. The radius $r$ is fixed, the speed $v$ is constant, and the angular velocity $\omega$ is constant. The period $T$, which is the time for one full revolution, is also constant, as is the frequency $f$, the number of revolutions per second.

What changes is the direction of the velocity vector. At every point on the circle, the velocity is tangent to the path. Because velocity is a vector, a change in direction means a change in velocity, even if the speed does not change.

In uniform circular motion, constant speed does not mean zero acceleration. The velocity changes direction, so the object is accelerating.

Geometry of the motion

For one complete revolution, the object travels a distance equal to the circumference of the circle,

$$
\text{distance in one revolution} = 2\pi r
$$

If the time for one revolution is $T$, then the speed is

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

Since frequency is the number of revolutions per second,

$$
f = \frac{1}{T}
$$

so the speed can also be written as

$$
v = 2\pi r f
$$

Angular velocity is related to period and frequency by

$$
\omega = \frac{2\pi}{T} = 2\pi f
$$

Using this, the speed becomes

$$
v = r\omega
$$

Important formulas for uniform circular motion:
$$
v = \frac{2\pi r}{T}, \qquad f = \frac{1}{T}, \qquad \omega = \frac{2\pi}{T} = 2\pi f, \qquad v = r\omega
$$

Velocity direction

Although the object returns again and again to the same circular path, its velocity vector is never fixed in direction. At the top of the circle it points one way, at the side another way, and so on. It is always tangent to the circle.

This changing direction is the key idea behind uniform circular motion. An inward acceleration is needed to keep turning the velocity vector toward the center.

Rendering scientific sketch…

Velocity in uniform circular motion

Relation to angular motion

Uniform circular motion connects linear motion and rotational motion. If the object sweeps out equal angles in equal times, then its angular velocity is constant. In one full revolution, the angular displacement is $2\pi$ radians.

The linear speed depends on both the angular velocity and the distance from the center. A point farther from the center moves faster than a point closer in, even if both rotate with the same angular velocity.

For example, on a spinning disk, all points have the same $\omega$, but points near the edge have larger $v$ because $v = r\omega$.

Centripetal idea

To move in a circle, the object must continuously turn inward. This inward turning is described by centripetal acceleration, which points toward the center of the circle. In uniform circular motion, this acceleration has constant magnitude because both $v$ and $r$ are constant.

The detailed formula for centripetal acceleration belongs to the next topic, but the main idea is already clear here. Uniform circular motion always requires an inward acceleration.

Uniform circular motion requires acceleration toward the center of the circle. Without this inward acceleration, the object would move off in a straight line.

Rendering scientific sketch…

Velocity tangent and inward acceleration

Comparing key quantities

Quantity	Meaning	Constant in uniform circular motion?
$r$	Radius of circle	Yes
$v$	Speed	Yes
$\vec v$	Velocity vector	No, direction changes
$\omega$	Angular velocity	Yes
$T$	Period	Yes
$f$	Frequency	Yes
$\vec a$	Acceleration vector	No, direction changes as object moves

A simple example

Suppose a car moves around a circular track of radius $50 \, \text{m}$ and completes one lap in $20 \, \text{s}$. Its speed is

$$
v = \frac{2\pi r}{T} = \frac{2\pi(50)}{20} = 5\pi \, \text{m/s}
$$

$$
v \approx 15.7 \, \text{m/s}
$$

Its frequency is

$$
f = \frac{1}{20} = 0.05 \, \text{Hz}
$$

and its angular velocity is

$$
\omega = \frac{2\pi}{20} = \frac{\pi}{10} \, \text{rad/s}
$$

These values remain constant as long as the motion stays uniform.

Physical meaning

Uniform circular motion is one of the simplest examples where motion can be steady in one sense and changing in another. The speed is steady, but the velocity is not. This teaches an important lesson in physics, vectors matter. Motion is not described fully by speed alone.

It also prepares the way for many later ideas in mechanics, especially forces in circular motion, rotating systems, planetary motion, and waves. Here the central point is simple: an object in uniform circular motion moves around a circle at constant speed while its direction, velocity, and acceleration continuously change.

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