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SM_chapter10 - 10 Rotation of a Rigid Object About a Fixed...

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10 Rotation of a Rigid Object About a Fixed Axis CHAPTER OUTLINE 10.1 Angular Position, Velocity, and Acceleration 10.2 Rotational Kinematics: Rotational Motion with Constant Angular Acceleration 10.3 Angular and Translational Quantities 10.4 Rotational Energy 10.5 Calculation of Moments of Inertia 10.6 Torque 10.7 Relationship Between Torque and Angular Acceleration 10.8 Work, Power, and Energy in Rotational Motion 10.9 Rolling Motion of a Rigid Object ANSWERS TO QUESTIONS Q10.1 1 rev min, or π 30 rad s. The direction is horizontally into the wall to represent clockwise rotation. The angular velocity is constant so α = 0. Q10.2 The vector angular velocity is in the direction + ˆ k . The vector angular acceleration has the direction ˆ k . *Q10.3 The tangential acceleration has magnitude (3 s 2 ) r where r is the radius. It is constant in time. The radial acceleration has magnitude ω 2 r , so it is (4 s 2 ) r at the first and last moments mentioned and it is zero at the moment the wheel reverses. Thus we have b = f > a = c = e > d = 0. *Q10.4 (i) answer (d). 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 . (ii) answer (c). If the driver uses the gearshift and the gas pedal to keep the tachometer readings and the air speeds comparable before and after the tire switch, there should be no effect. *Q10.5 (i) answer (a). Smallest I is about x axis, along which the larger-mass balls lie. (ii) answer (c). The balls all lie at a distance from the z axis, which is perpendicular to both the x and y axes and passes through the origin. Q10.6 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. *Q10.7 The accelerations are not equal, but greater in case (a). The string tension above the 5.1-kg object is less than its weight while the object is accelerating down. Q10.8 You could measure the time that it takes the hanging object, of known mass m , to fall a measured distance after being released from rest. Using this information, the linear acceleration of the mass can be calculated, and then the torque on the rotating object and its angular acceleration. 245 FIG. Q10.1 13794_10_ch10_p245-282.indd 245 13794_10_ch10_p245-282.indd 245 1/4/07 12:05:18 PM 1/4/07 12:05:18 PM
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*Q10.9 answers (a), (b), and (e). The object must rotate with nonzero angular acceleration. The center of mass can be constant in location if it is on the axis of rotation. Q10.10 You could use ω α = t and v = at . The equation v = R ω is valid in this situation since a R = α . Q10.11 The angular speed ω would decrease. The center of mass is farther from the pivot, but the moment of inertia increases also. *Q10.12 answer (f ). The sphere of twice the radius has eight times the volume and eight times the mass.
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