SM_PDF_chapter10 - Rotational Motion CHAPTER OUTLINE 10.1...

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257 Rotational Motion CHAPTER OUTLINE 10.1 Angular Position, Speed, and Acceleration 10.2 Rotational Kinematics The Rigid Object Under Constant Angular Acceleration 10.3 Relations Between Rotational and Translational Quantities 10.4 Rotational Kinetic Energy 10.5 Torque and the Vector Product 10.6 The Rigid Object in Equilibrium 10.7 The Rigid Object Under a Net Torque 10.8 Angular Momentum 10.9 Conservation of Angular Momentum 10.10 Precessional Motion of Gyroscopes 10.11 Rolling Motion of Rigid Objects 10.12 Context Connection Turning the Spacecraft ANSWERS TO QUESTIONS Q10.1 1 rev/min, or π 30 rad/s. Into the wall (clockwise rotation). α = 0. FIG. Q10.1 Q10.2 The speedometer will be inaccurate. The speedometer measures the number of revolutions per second of the tires. A larger tire will travel more distance in one full revolution as 2 π r . Q10.3 The object will start to rotate if the two forces act along different lines. Then the torques of the forces will not be equal in magnitude and opposite in direction. Q10.4 A quick flip will set the hard–boiled egg spinning faster and more smoothly. The raw egg loses mechanical energy to internal fluid friction. Q10.5 No, only if its angular momentum changes. Q10.6 I MR CM = 2 , I MR CM = 2 , I MR = 1 3 2 , I MR CM = 1 2 2 Q10.7 Since the source reel stops almost instantly when the tape stops playing, the friction on the source reel axle must be fairly large. Since the source reel appears to us to rotate at almost constant angular velocity, the angular acceleration must be very small. Therefore, the torque on the source reel due to the tension in the tape must almost exactly balance the frictional torque. In turn, the frictional torque is nearly constant because kinetic friction forces don’t depend on velocity, and the radius of the axle where the friction is applied is constant. Thus we conclude that the torque exerted by the tape on the source reel is essentially constant in time as the tape plays. continued on next page
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258 Rotational Motion As the source reel radius R shrinks, the reel’s angular speed ω = v R must increase to keep the tape speed v constant. But the biggest change is to the reel’s moment of inertia. We model the reel as a roll of tape, ignoring any spool or platter carrying the tape. If we think of the roll of tape as a uniform disk, then its moment of inertia is I MR = 1 2 2 . But the roll’s mass is proportional to its base area π R 2 . Thus, on the whole the moment of inertia is proportional to R 4 . The moment of inertia decreases very rapidly as the reel shrinks! The tension in the tape coming into the read-and-write heads is normally dominated by balancing frictional torque on the source reel, according to TR τ friction . Therefore, as the tape plays the tension is largest when the reel is smallest. However, in the case of a sudden jerk on the tape, the rotational dynamics of the source reel becomes important. If the source reel is full, then the moment of inertia, proportional to R 4 , will be so large that higher tension in the tape will be required to give the source reel its angular acceleration. If the reel is nearly empty, then the same tape acceleration
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