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Unformatted text preview: 1 2 Chapter 10 Rotational Motion About a Fixed Axis 3 101 Angular Quantities l = R θ = θ = 1 rad Circle = 360 o = 2 π rad = 1 rev l R Radian is defined as the angle subtended by an arc whose length is equal to the radius. 4 l = R θ 5 R = perpendicular distance from axis. 6 Example 101 Birds of prey—in radians. A particular bird’s eye can just distinguish objects that subtend an angle no smaller that about 3 x 104 rad. (a) How many degrees is this? (b) How small an object can the bird just distinguish when flying at a height of 100 m? 7 Derivation of Angular Quantities 8 9 10 11 Tangential velocity 12 13 Frequency • Relating angular velocity to the frequency, f , ( ν ) of rotation, where frequency means the number of revolutions per second. • One revolution corresponds to 2 π radians. • 1 rev/s = 2 π rad/s. 1 Hz (Hertz) = 1 rev/s. ϖ 2 π f = ; ϖ = 2 π f 14 Example 102 Hard drive. The platter of a hard disk of a computer rotates at 5400 rpm (revolutions per minute). ( a ) What is the angular velocity of the disk? ( b ) If the reading head of the drive is located 3.0 cm from the rotation axis, what is the speed of the disk below it? ( c ) What is the linear acceleration of this point? ( d ) If a single bit requires 5.0 μ m of length along the motion of direction, how many bits per second can the writing head write when it is 3.0 cm from the axis? ( e ) If the disk took 3.6 s to spin up to 5400 rpm from rest, what was the average acceleration? 15 102 Kinematic Equations for Uniformly Accelerated Rotational Motion Angular θ = ϖ ο t + ½ α t 2 ϖ = ϖ ο + α t ϖ 2 = ϖ ο 2 + 2 αθ Linear x = v o t + ½ at 2 v = v o + at v 2 = v o 2 + 2 ax These equations are valid only for constant a and α . 16 Example 103 Hard drive again. Through how many revolutions did the hard drive in example 102 turn to reach 5400 rpm during its acceleration period? Assume a constant angular acceleration. 17 103 Rolling Motion (without slipping) • Rolling without slipping depends on static friction between the wheel, ball, etc. and the ground. • The friction is static because the rolling object’s point of contact with the ground is at rest at each moment. • Kinetic friction comes in if the object skids, that is, slides. VERY IMPORTANT CONCEPT! 18 Static friction: wheel does not slide Center of mass (CM) Rolling Without Slipping Involves Both Rotation and Translation. Reference frame attached to ground Reference frame attached to wheel 19 Reference frame attached to the ground 20 Reference frame attached to the wheel 21 Example 104 Bicycle. A bicycle slows down uniformly from v o = 8.40 m/s to rest over a distance of 115 m. Each wheel and tire has an overall diameter of 68.0 cm. Determine ( a ) the angular velocity of the wheels at the initial instant ( t = 0), ( b ) the total number of revolutions each wheel rotates before coming to rest, ( c ) the angular acceleration of the wheel, and ( d ) the time it took to come to a stop. 22 104...
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 Summer '09
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