Module 5 -- Uniform Circular Motion

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Up until now in this Unit we have been specifying details of the acceleration experienced by an object (e.g. constant acceleration, zero acceleration) and then using the concepts of mechanics to learn about the motion. In this Module, we will work the other way: we will specify a motion we are interested in and then use the concepts of mechanics to learn about the acceleration that is being experienced. The motion we are interested in is motion in a circle of constant radius at constant speed, which is referred to as uniform circular motion.

Learning Goals

After working through this module, you should be able to:

• Explain how it is possible that an object moving with constant speed can be undergoing an acceleration.
• Define centripetal acceleration.
• Draw free body diagrams for objects executing uniform circular motion.

Observing the Motion using the Physics Teaching Technology Resource

In order to describe a motion, we have to first observe it carefully. One of the great advantages of mechanics over other branches of physics is the fact that we can use our eyes to make the observation (this is not true when studying electricity or quantum mechanics, for example). The Physics and Astronomy Education Research Group at Rutgers University have created a library of simple video experiments to illustrate many kinds of motion. Below, we present three of their videos that involve circular motion. Observe them carefully. Particularly, think about the forces acting on the person or object that is moving in a circle in each video.

These videos are part of the Physics Teaching Technology Resource
© Rutgers University, used with permission.

Please wait while the movies load up. (This may take a while.)
You will need QuickTime to view the movies.

Centripetal Acceleration

We are now ready to discuss the properties of the acceleration associated with uniform circular motion. There are three important claims we will make about this acceleration.

Claim 1: An object executing uniform circular motion must be experiencing an interaction.

This claim is based upon Newton's First Law, which says that an object that is not experiencing any interaction will necessarily move in a straight line. You can see the evidence for this claim in the video segments above. The bowling ball in the first video moves in a straight line unless acted upon by the hammer. In the second video segment, Professor Etkina skates in a straight line unless acted upon by the rope (first circular path) or by friction from the ground (second circular path). In the third video segment, the ball only moves in a circular path when constrained to do so by the interaction from the wall. When the wall is removed, the ball travels a straight path.

Claim 2: The net interaction on an object in uniform circular motion must produce a force pointing toward the center of the circular path.

The evidence for this claim can be seen by careful observation of the videos using what we have already learned about the properties of interactions. In the first video segment, the hammer is always applied to the bowling ball on the side directly opposite the center of the circle. If we assume that the ball and hammer are fairly smooth (so there is little friction) then the contact interaction points toward the center of the circle. In the second video, the rope turning Professor Etkina clearly pulls her toward the center of the circle when she first makes a circular path. In her second circular path, she kicks out away from the center, producing a reaction force that (according to Newton's Third Law) points inward toward the center. In the third video segment, the normal force applied by the wall is perpendicular to the wall and therefore points toward the center of the circle (again, we assume that friction is small).

Claim 3: The size of the object's acceleration is given by v2/r, where v is the speed of the object and r is the radius of the circular path.

Evidence for our final claim cannot be observed directly in the video, but must be derived using what we have assumed about the motion (uniform speed and uniform radius). Below, we present two possible derivations. One uses trigonometry plus limits, and the other uses calculus.

The acceleration associated with circular motion is generally called centripetal acceleration because of fact 2. "Centripetal" means "center-seeking" and is meant to remind us that the acceleration of an object moving in a circle constantly changes direction so as to always point at the center of the circular path.

The Uniform Circular Motion (Centripetal Acceleration) Model

We are now ready to introduce the Uniform Circular Motion (Centripetal Acceleration) Model. In principle, this Model involves a full description of the position, velocity and acceleration of an object moving in a circle of constant radius with constant speed. In practice, the most important thing to remember about Uniform Circular Motion is that any object executing this motion must be experiencing a centripetal acceleration that obeys the three facts outlined above. By adding in what we have learned about Newton's Laws, we can see that if we are given a problem involving an object executing Uniform Circular Motion we have important information about the magnitude and direction of the net force on this object (since the net force is equal to ma, and we know a = v2/r and that it points toward the center of the circle).

Illustrative Example: The International Space Station

By navigating to the site http://spaceflight.nasa.gov/realdata/tracking/ you can obtain tracking information giving the altitude and speed of the International Space Station (ISS). Suppose that the information displayed on the site was as shown in the screen capture above. By making the assumption that the space station's orbit is a circle with its center at the center of the earth, find the approximate magnitude of the acceleration experienced by the space station as a result of the gravitational pull of the earth.