# Introduction to Uniform Circular Motion

Did you ever stop to count the number of controls that cause acceleration in your car? At a minimum, you have a gas pedal, a brake, and a steering wheel. That's right, the steering wheel produces an acceleration on your car, because it causes you to change direction which changes your velocity, even if it doesn't change your speed. That's what Uniform Circular Motion is all about - the changing of the velocity of an object as it travels in a circle at a constant speed. In this lesson, we want to examine the relationship a little further between linear and circular motion and develop a concept call Centripetal Acceleration.

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Let's start by learning some common terms (or definitions, if you will):

Velocity – a vector quantity whose magnitude is a body's speed and whose direction is the body's direction of motion (i.e. speed with direction.)

Acceleration – The rate of change of velocity per unit time

Uniform Circular Motion – an object moving in a circle at a constant speed.

Centripetal Acceleration – does not change speed, only changes direction, also called radial acceleration. (Symbol = “Ar” or “Ac”)

Center Seeking – always toward the center of the circle.

Frequency – The number of complete cycles of a periodic process occurring per unit time. (Per second) (Symbol = “f”)

Period – the time it takes the object to complete one cycle or revolution. (Symbol = “T”) Usually expressed in seconds, but may not always be convenient to do so.  For example, the period of the Earth around the Sun is 31,557,600 seconds, but it may be more convenient to call this one year.

Consider an object that moves in a circle.Newton's1st Law says that the object will move in a straight line at a constant speed unless acted upon by an outside force.We'll consider that force in a later lesson.For now, let's go back to our basics:acceleration is a change in velocity per unit time.At time t1, our object has velocity v1. At time t2,the velocity has changed to that shown.If we do vector subtraction ofv2- v1, we find a resultant vector that always pointstowards the center of the circle.This is our "center seeking" vector. Our acceleration is towards the center, and is given by the formula

ar= v²/r      .

We call this acceleration "Centripetal Acceleration".

The vector answer will always be pointing in towards the center because it is center seeking, which means that the acceleration is also pointing towards the center.

Also, acceleration and velocity do not necessarily have to be in the same direction. In fact, in UCM, they will be  at 90o angles to each other.

A simple example problem for Uniform Circular Motion:

An object is rotating in a circle of radius = 4m. The object makes 4 revolutions in two seconds. What is its centripetal acceleration:

ar=V²/r =(2πr/T)²/r

T = .5 sec, so

ar = (2π4m/.5s)²/4m = 631.6 m/s² towards the center

Often it is helpful to see what someone else has to say on the topic. Be sure to check out:

http://www.physicsclassroom.com/Class/circles/u6l1a.html

http://www.physicsclassroom.com/Class/circles/U6L1b.html

For Practice Problems, Try:

http://imagine.gsfc.nasa.gov/YBA/cyg-X1-mass/circular-quiz.html

From the University of Oregon:

Calculating Centrepital Acceleration

Giancoli Multiple Choice Practice Questions (Select "Practice Problems) and try some. Don't worry - you won't be able to do all of them yet.