Física para ingenieros

10
Rotation of a Rigid Object About a Fixed Axis
CHAPTER OUTLINE
10.1 10.2 Angular Position, Velocity, and Acceleration Rotational Kinematics: Rotational Motion with Constant Angular Acceleration Angular and Translational Quantities Rotational Energy Calculation of Moments of Inertia Torque Relationship Between Torque and Angular Acceleration Work, Power, and Energy in Rotational MotionRolling Motion of a Rigid Object

ANSWERS TO QUESTIONS
Q10.1 1 rev min, or

π rad s. The direction is horizontally into 30 the wall to represent clockwise rotation. The angular
velocity is constant so α = 0.

10.3 10.4 10.5 10.6 10.7 10.8 10.9

FIG. Q10.1

Q10.2

ˆ The vector angular velocity is in the direction + k. The vector angular acceleration has the ˆ direction − k.

*Q10.3 Thetangential acceleration has magnitude (3 s2)r where r is the radius. It is constant in time. The radial acceleration has magnitude ω 2r, so it is (4 s2)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 willtravel more distance in one full revolution as 2π r. 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. answer (a). Smallest I is about x axis, along which the larger-mass balls lie. answer (c). The balls all lie at a distance from the z axis, which is perpendicularto both the x and y axes and passes through the origin.

(ii)

*Q10.5 (i) (ii)

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 weightwhile 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.
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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 = Rω is valid in this situation since a = Rα . Q10.11 The angular speed ω would decrease. The center of mass is farther from the pivot, but themoment of inertia increases also.

*Q10.12 answer (f ). The sphere of twice the radius has eight times the volume and eight times the mass. Then r 2 in I = (2 5) mr 2 also gets four times larger. Q10.13 The moment of inertia depends on the distribution of mass with respect to a given axis. If the axis is changed, then each bit of mass that makes up the object is a different distance from the axis.In an example in section 10.5 in the text, the moment of inertia of a uniform rigid rod about an axis perpendicular to the rod and passing through the center of mass is derived. If you spin a pencil back and forth about this axis, you will get a feeling for its stubbornness against changing rotation. Now change the axis about which you rotate it by spinning it back and forth about the axis thatgoes down the middle of the graphite. Easier, isn’t it? The moment of inertia about the graphite is much smaller, as the mass of the pencil is concentrated near this axis. Q10.14 A quick flip will set the hard–boiled egg spinning faster and more smoothly. Inside the raw egg, the yolk takes some time to start rotating. The raw egg also loses mechanical energy to internal fluid friction. Q10.15 1 1...