1. Definition

Uniform Circular Motion (UCM) is the motion of an object traveling in a circular path at a constant speed.

Although the speed remains constant, the object is still accelerating because its direction of motion continuously changes.

2. Characteristics of Uniform Circular Motion

In UCM:

• the path is circular
• the speed is constant
• the velocity changes direction continuously
• the acceleration points toward the center of the circle

Because velocity changes, Newton’s Second Law requires a net force.

3. Velocity in Circular Motion

The velocity vector is always:

• tangent to the circular path
• perpendicular to the radius

As the object moves around the circle, the direction of velocity changes continuously.

4. Centripetal Acceleration

The acceleration responsible for changing the direction of motion is called centripetal acceleration.

It always points toward the center of the circle.

where:

• \(a_{c}\)​ = centripetal acceleration
• \(v\) = speed
• \(r\) = radius of the circular path

5. Centripetal Force

According to Newton’s Second Law:

\(\sum F=ma\)

For circular motion:

\(F_c=ma_c\)​

Substituting the centripetal acceleration:

\(F_c=\frac{mv^2}{r}\)​

This is called the centripetal force requirement.

6. Important Concept

Centripetal force is not a new type of force.

Instead, it is the name given to the net force directed toward the center.

Possible sources include:

• tension
• gravity
• friction
• normal force
• combinations of these forces

7. Example: Object on a String

An object moves in a horizontal circle attached to a string.

The tension provides the centripetal force:

\(T=\frac{mv^2}{r}\)​

The string continuously pulls the object toward the center.

8. Example: Car Turning on a Flat Road

A car traveling around a curve requires a centripetal force.

This force is supplied by static friction:

\(f_s=\frac{mv^2}{r}\)​

Without sufficient friction, the car slides outward.

9. Example: Satellite Orbit

For a satellite orbiting Earth:

Gravity supplies the centripetal force.

\(F_g=\frac{mv^2}{r}\)​

The satellite is constantly falling toward Earth while moving forward fast enough to remain in orbit.

10. Angular Speed

Angular speed describes how quickly the object rotates.

\(\omega=\frac{\Delta\theta}{\Delta t}\)​

where:

• $\omega$ = angular speed
• $\theta$ = angular displacement

Relationship between linear and angular speed:

\(v=r\omega\)

11. Period and Frequency

Period

Time for one complete revolution.

\(T=\frac{2\pi r}{v}\)​

Frequency

Number of revolutions per second.

\(f=\frac{1}{T}\)​

12. Common Misconception

Students often think an object moving at constant speed has no acceleration.

In circular motion:

• speed is constant
• velocity is not constant

Therefore:

\(a_c\neq0\)

A net force toward the center is always required.

13. Free-Body Diagram Strategy

For circular-motion problems:

Step 1

Draw all real forces.

Step 2

Identify the force(s) directed toward the center.

Step 3

Apply Newton’s Second Law radially:

\(\sum F_{toward center}=\frac{mv^2}{r}\)​

Summary

Uniform Circular Motion is motion at constant speed along a circular path.

Centripetal acceleration:

\(a_c=\frac{v^2}{r}\)​

Centripetal force:

\(F_c=\frac{mv^2}{r}\)​

Key ideas:

• velocity is tangent to the circle
• acceleration points toward the center
• a net inward force is required
• centripetal force is provided by real forces such as tension, gravity, friction, or normal force