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# 10 rotation of a rigid object about a fixed axis

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### 10 rotation of a rigid object about a fixed axis

1. 1. P U Z Z L E RDid you know that the CD inside thisplayer spins at different speeds, depend-ing on which song is playing? Why wouldsuch a strange characteristic be incor-porated into the design of every CDplayer? (George Semple)c h a p t e r Rotation of a Rigid Object About a Fixed Axis Chapter Outline 10.1 Angular Displacement, Velocity, 10.5 Calculation of Moments of and Acceleration Inertia 10.2 Rotational Kinematics: Rotational 10.6 Torque Motion with Constant Angular 10.7 Relationship Between Torque Acceleration and Angular Acceleration 10.3 Angular and Linear Quantities 10.8 Work, Power, and Energy in 10.4 Rotational Energy Rotational Motion292
3. 3. 294 CHAPTER 10 Rotation of a Rigid Object About a Fixed Axis y Q ,tf r P, ti θf In a short track event, such as a 200-m or θi 400-m sprint, the runners begin from stag- x gered positions on the track. Why don’t O they all begin from the same line?Figure 10.2 A particle on a rotat-ing rigid object moves from P to Qalong the arc of a circle. In thetime interval ⌬t ϭ t f Ϫ t i , the ra- As the particle in question on our rigid object travels from position P to positiondius vector sweeps out an angle Q in a time ⌬t as shown in Figure 10.2, the radius vector sweeps out an angle ⌬␪ ϭ ␪f⌬␪ ϭ ␪f Ϫ ␪i . Ϫ ␪i . This quantity ⌬␪ is deﬁned as the angular displacement of the particle: ⌬␪ ϭ ␪f Ϫ ␪i (10.2) We deﬁne the average angular speed ␻ (omega) as the ratio of this angular dis- placement to the time interval ⌬t: ␪f Ϫ ␪i ⌬␪ Average angular speed ␻ϵ ϭ (10.3) tf Ϫ ti ⌬t In analogy to linear speed, the instantaneous angular speed ␻ is deﬁned as the limit of the ratio ⌬␪/⌬t as ⌬t approaches zero: ⌬␪ d␪ Instantaneous angular speed ␻ ϵ lim ϭ (10.4) ⌬t:0 ⌬t dt Angular speed has units of radians per second (rad/s), or rather secondϪ1 (sϪ1) because radians are not dimensional. We take ␻ to be positive when ␪ is in- creasing (counterclockwise motion) and negative when ␪ is decreasing (clockwise motion). If the instantaneous angular speed of an object changes from ␻i to ␻f in the time interval ⌬t, the object has an angular acceleration. The average angular ac- celeration ␣ (alpha) of a rotating object is deﬁned as the ratio of the change in the angular speed to the time interval ⌬t : ␻f Ϫ ␻i ⌬␻ Average angular acceleration ␣ϵ ϭ (10.5) tf Ϫ ti ⌬t
4. 4. 10.1 Angular Displacement, Velocity, and Acceleration 295 ω Figure 10.3 The right-hand rule for deter- ω mining the direction of the angular velocity vector. In analogy to linear acceleration, the instantaneous angular acceleration isdeﬁned as the limit of the ratio ⌬␻/⌬t as ⌬t approaches zero: ⌬␻ d␻ ␣ ϵ lim ϭ (10.6) Instantaneous angular ⌬t:0 ⌬t dt accelerationAngular acceleration has units of radians per second squared (rad/s2 ), or just sec-ondϪ2 (sϪ2 ). Note that ␣ is positive when the rate of counterclockwise rotation isincreasing or when the rate of clockwise rotation is decreasing. When rotating about a ﬁxed axis, every particle on a rigid object rotatesthrough the same angle and has the same angular speed and the same an-gular acceleration. That is, the quantities ␪, ␻, and ␣ characterize the rotationalmotion of the entire rigid object. Using these quantities, we can greatly simplifythe analysis of rigid-body rotation. Angular position (␪), angular speed (␻), and angular acceleration (␣) areanalogous to linear position (x), linear speed (v), and linear acceleration (a). Thevariables ␪, ␻, and ␣ differ dimensionally from the variables x, v, and a only by afactor having the unit of length. We have not speciﬁed any direction for ␻ and ␣. Strictly speaking, thesevariables are the magnitudes of the angular velocity and the angular accelera-tion vectors ␻ and ␣, respectively, and they should always be positive. Becausewe are considering rotation about a ﬁxed axis, however, we can indicate the di-rections of the vectors by assigning a positive or negative sign to ␻ and ␣, as dis-cussed earlier with regard to Equations 10.4 and 10.6. For rotation about a ﬁxedaxis, the only direction that uniquely speciﬁes the rotational motion is the di-rection along the axis of rotation. Therefore, the directions of ␻ and ␣ arealong this axis. If an object rotates in the xy plane as in Figure 10.1, the direc-tion of ␻ is out of the plane of the diagram when the rotation is counterclock-wise and into the plane of the diagram when the rotation is clockwise. To illus-trate this convention, it is convenient to use the right-hand rule demonstrated inFigure 10.3. When the four ﬁngers of the right hand are wrapped in the direc-tion of rotation, the extended right thumb points in the direction of ␻. The di-rection of ␣ follows from its deﬁnition d␻/dt. It is the same as the direction of␻ if the angular speed is increasing in time, and it is antiparallel to ␻ if the an-gular speed is decreasing in time.
6. 6. 10.3 Angular and Linear Quantities 297 TABLE 10.1 Kinematic Equations for Rotational and Linear Motion Under Constant Acceleration Rotational Motion About a Fixed Axis Linear Motion ␻f ϭ ␻ i ϩ ␣t vf ϭ vi ϩ at ␪f ϭ ␪i ϩ ␻ i t ϩ 1 2 ␣t 2 xf ϭ xi ϩ v i t ϩ 1 2 at 2 ␻f ϭ ␻ i ϩ 2␣(␪f Ϫ ␪i ) 2 2 vf ϭ v i ϩ 2a(xf Ϫ xi ) 2 2 y10.3 ANGULAR AND LINEAR QUANTITIESIn this section we derive some useful relationships between the angular speed and vacceleration of a rotating rigid object and the linear speed and acceleration of an Parbitrary point in the object. To do so, we must keep in mind that when a rigid ob-ject rotates about a ﬁxed axis, as in Figure 10.4, every particle of the object moves rin a circle whose center is the axis of rotation. θ We can relate the angular speed of the rotating object to the tangential speed x Oof a point P on the object. Because point P moves in a circle, the linear velocityvector v is always tangent to the circular path and hence is called tangential velocity.The magnitude of the tangential velocity of the point P is by deﬁnition the tangen-tial speed v ϭ ds/dt, where s is the distance traveled by this point measured along Figure 10.4 As a rigid object ro-the circular path. Recalling that s ϭ r ␪ (Eq. 10.1a) and noting that r is constant, tates about the ﬁxed axis throughwe obtain O, the point P has a linear velocity v that is always tangent to the circu- ds d␪ vϭ ϭr lar path of radius r. dt dtBecause d␪/dt ϭ ␻ (see Eq. 10.4), we can say Relationship between linear and v ϭ r␻ (10.10) angular speedThat is, the tangential speed of a point on a rotating rigid object equals the per-pendicular distance of that point from the axis of rotation multiplied by the angu-lar speed. Therefore, although every point on the rigid object has the same angu-lar speed, not every point has the same linear speed because r is not the same forall points on the object. Equation 10.10 shows that the linear speed of a point onthe rotating object increases as one moves outward from the center of rotation, aswe would intuitively expect. The outer end of a swinging baseball bat moves muchfaster than the handle. QuickLab We can relate the angular acceleration of the rotating rigid object to the tan- Spin a tennis ball or basketball andgential acceleration of the point P by taking the time derivative of v: watch it gradually slow down and stop. Estimate ␣ and at as accurately dv d␻ as you can. at ϭ ϭr dt dt Relationship between linear and at ϭ r ␣ (10.11) angular accelerationThat is, the tangential component of the linear acceleration of a point on a rotat-ing rigid object equals the point’s distance from the axis of rotation multiplied bythe angular acceleration.