Concept Summary

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Topic | Discussion | Learning Aids
| 5.1 UNIFORM CIRCULAR MOTION Uniform circular motion is the motion of an object traveling at a constant (uniform) speed on a circular path. |
Period of the motion | The period T is the time required for the object to travel once around the circle. The speed v of the object is related to the period and the radius r of the circle by

| (5.1)

| 5.2 CENTRIPETAL ACCELERATION An object in uniform circular motion experiences an acceleration, known as centripetal acceleration. The magnitude ac of the centripetal acceleration is |
Centripetal acceleration (magnitude) |

| (5.2)

Centripetal acceleration (direction) | where v is the speed of the object and r is the radius of the circle. The direction of the centripetal acceleration vector always points toward the center of the circle and continually changes as the object moves. |
| 5.3 CENTRIPETAL FORCE To produce a centripetal acceleration, a net force pointing toward the center of the circle is required. This net force is called the centripetal force, and its magnitude Fc is |
Centripetal force (magnitude) |

| (5.3)

Centripetal force (direction) | where m and v are the mass and speed of the object, and r is the radius of the circle. The direction of the centripetal force vector, like that of the centripetal acceleration vector, always points toward the center of the circle. |

Use Self-Assessment Test 5.1 to evaluate your understanding of Sections 5.1–5.3.

| 5.4 BANKED CURVES A vehicle can negotiate a circular turn without relying on static friction to provide the centripetal force, provided the turn is banked at an angle relative to the horizontal. The angle at which a friction-free curve must be banked is related to the speed v of the vehicle, the radius r of the curve, and the magnitude g of the acceleration due to gravity by |
Angle of a banked turn |

| (5.4)

| 5.5 SATELLITES IN CIRCULAR ORBITS When a satellite orbits the earth, the gravitational force provides the centripetal force that keeps the satellite moving in a circular orbit. The speed v and period T of a satellite depend on the mass ME of the earth and the radius r of the orbit according to |
Orbital speed |

| (5.5)

Orbital period |

| (5.6)

where G is the universal gravitational constant.
| 5.6 APPARENT WEIGHTLESSNESS AND ARTIFICIAL GRAVITY The apparent weight of an object is the force that it exerts on a scale with which it is in contact. All objects, including people, on board an orbiting satellite are in free-fall, since they experience negligible air resistance and they have an acceleration that is equal to the acceleration due to gravity. When a person is in freefall, his or her apparent weight is zero, because both the person and the scale fall freely and cannot push against one another. |
| 5.7 VERTICAL CIRCULAR MOTION Vertical circular motion occurs when an object, such as a motorcycle, moves on a vertical circular path. The speed of the object often varies from moment to moment, and so do the magnitudes of the centripetal acceleration and centripetal force. |

Use Self-Assessment Test 5.2 to evaluate your understanding of Sections 5.4–5.7.

Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.