Chapter 7 Skill Building

Acceleration is the time derivative of velocity

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Acceleration is the time derivative of velocity. Because velocity is a vector, it can change in two ways: the length (magnitude) can change and/or the direction can change. The latter type of change has a special name, the centripetal acceleration . In this problem we consider a mass moving in a circle of radius with angular velocity , . The main point of the problem is to compute the acceleration using geometric arguments. Part A What is the velocity of the mass at a time ? You can work this out geometrically with the help of the hints, or by differentiating the expression for given in the introduction. Part A.1 Part not displayed Part A.2 Part not displayed Express this velocity in terms of , , , and the unit vectors and . ANSWER: = Assume that the mass has been moving along its circular path for some time. You start timing its (4 of 16)4/11/2006 2:52:13 PM
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MasteringPhysics: Assignment Print View motion with a stopwatch when it crosses the positive x axis, an instant that corresponds to . [Notice that when , .] For the remainder of this problem, assume that the time is measured from the moment you start timing the motion. Then the time refers to the moment a time before you start your stopwatch. Part B What is the velocity of the mass at a time ? Express this velocity in terms of , , , and the unit vectors and . ANSWER: = Part C What is the average acceleration of the mass during the time interval from to ? Hint C.1 Hint not displayed Express this acceleration in terms of , , , and the unit vectors and . ANSWER: = Part D (5 of 16)4/11/2006 2:52:13 PM
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MasteringPhysics: Assignment Print View What is the magnitude of this acceleration in the limit of small ? In this limit, the average acceleration becomes the instantaneous acceleration. Part D.1 Part not displayed Express your answer in terms of and . ANSWER: = Part E Consider the following statements: a. The centripetal acceleration might better be expressed as because it is a vector. b. The magnitude of the centripetal acceleration is . c. The magnitude of the centripetal acceleration is . d. A particle that is going along a path with local radius of curvature at speed experiences a centripetal acceleration . e. If you are in a car turning left, the force you feel pushing you to the right is the force that causes the centripetal acceleration. In these statements refers to the component of the velocity of an object in the direction toward or away from the origin of the coordinate system or the rotation axis. Conversely, refers to the component of the velocity perpendicular to . Identify the statement or statements that are false . ANSWER: a only b only c only d only e only b and e c and e d and e That's right; the true statements are therefore: a. The centripetal acceleration might better be expressed as because it is a vector.
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