Chapter 12: Uniform Circular Motion

12.1 Introduction to Uniform Circular Motion (UCM)

Uniform circular motion describes the movement of an object traveling in a circular path with a constant speed. Although the speed remains constant, the object experiences acceleration due to the continuous change in its direction. This page will explore the basics of uniform circular motion.

12.2 Defining Uniform Circular Motion

In uniform circular motion, an object moves in a circular path with a constant speed. This means that the magnitude of its velocity remains unchanged. However, since the object’s direction is continuously changing, its velocity vector changes resulting in a non-zero acceleration. This acceleration is always directed towards the center of the circular path and is called the centripetal acceleration.

12.3 Centripetal Acceleration and Force

Centripetal acceleration is responsible for keeping the object moving in a circular path. It is calculated using the following formula:

where is the object’s linear speed (calculated simply as distance over time), and is the radius of the circular path.

Since there is acceleration involved, a force is also acting on the object, known as the centripetal force . According to Newton’s second law of motion, the centripetal force can be calculated as:

where is the mass of the object.

It is essential to note that centripetal force is not a separate force but rather a result of other forces acting on the object, such as tension, gravitational force, or friction, which keep the object in circular motion.

12.4 Key Concepts in Uniform Circular Motion

Period : The time taken for the object to complete one full revolution around the circular path.
Frequency : The number of complete revolutions the object makes in one second. Frequency is the reciprocal of the period, i.e., .
Angular Velocity : The rate at which the object moves around the circle, measured in radians per second. It can be calculated using the formula: .

12.5 Real-World UCM:

• Planetary motion: The planets orbiting the Sun move in approximately circular paths with nearly uniform motion.
• A car driving around a roundabout: The car maintains a constant speed while traveling around the roundabout, resulting in uniform circular motion.
• A spinning wheel: Each point on the wheel moves in a circular path with a constant speed.

Example:

You swing a rope above your head. The rope is 24 centimeters long. On one end is a ball of mass 2 kilograms. The ball makes 50 revolutions in 28 seconds. What is the average velocity of the ball? What is the centripetal acceleration? Finally, what is the cetripetal force?

Let’s start by determining the average velocity of the ball. One revolution means one full trip around the circle. Thus, a revolution starts and ends in the same place. Doing that fifty times doesn’t change anything. You still start and end in the same place. The displacement of the ball is 0 meters as shown below.

In order to solve for the centripetal acceleration, we need the average speed of the ball.

The distance traveled in one revolution is just the circumference of the circle. . Thus the distance is equal to because the ball goes around the circle fifty times. Divide this number by the total time recorded (28 seconds) and you have solved for the speed. Now just plug in and solve for using the equation .

From here, it is simple to solve for . Simply plug in your values to the equation from earlier: .

And thus you have your final answers.

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Continue to Chapter 13: Friction

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