You slip a DVD into a player, and it starts spinning. You could describe its motion by giving the speed and direction of each point on the disc. But it’s much easier just to say that the disc is rotating at 800 revolutions per minute (rpm). As long as the disc is a rigid body—one whose parts remain in fixed positions relative to one another—then that single statement suffices to describe the motion of the entire disc.
How You’ll Use It
■ Concepts of rotational motion that you learn here set the groundwork for the vector treatment of rotation in Chapter 11.
■ Angular velocity and rotational inertia will prove useful in the important quantity called angular momentum.
■ Appreciating the analogies between linear and angular motion will help you to understand the seemingly counterintuitive behaviors that result from the vector nature of rotational motion, from the precession of Earth’s axis to the stability of a bicycle.
What You’re Learning
■ Here you’ll learn the rotational analogs of quantities in one-dimensional motion: angular position, angular velocity, and angular acceleration.
■ You’ll be able to solve rotational- motion problems analogously to problems involving one-dimensional motion in Chapter 2.
■ You’ll learn about torque, the rotational analog of force, and rotational inertia, the analog of mass.
■ You’ll learn the rotational analog of Newton’s second law.
■ You’ll learn how to calculate rotational inertias by integration.
■ You’ll learn to couple rotational and linear motion, including the special case of rolling motion.
■ You’ll learn how to calculate rotational kinetic energy.
What You Know
■ You know how to describe motion in one dimension using the concepts of position, velocity, and acceleration.
■ You know how to relate position, velocity, and acceleration in the special case of one-dimensional motion with constant acceleration.
■ You understand Newton’s second law and how to apply it to one- dimensional motion.
■ You can describe circular motion and the associated centripetal acceleration.
Rotational Motion
For a given blade mass, how should you engineer a wind turbine’s blades so it’s easiest for the wind to get the turbine rotating?
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10.1 Angular Velocity and Acceleration 169
Angular Velocity
The rate at which a body rotates is its angular velocity—the rate at which the angular po- sition of any point on the body changes. With our 800-rpm DVD, the unit of angle was one full revolution (360°, or 2p radians), and the unit of time was the minute. But we could equally well express angular velocity in revolutions per second (rev/s), degrees per second 1°/s2, or radians per second (rad/s or simply s-1 since radians are dimensionless). Because of the mathematically simple nature of radian measure, we often use radians in calcula- tions involving rotational motion (Fig. 10.1).
We use the Greek symbol v (omega) for angular velocity and define average angular velocity v as
v = ∆u
∆t 1average angular velocity2 (10.1) where ∆u is the angular displacement—that is, the change in angular position— occurring in the time ∆t (Fig. 10.2). When angular velocity is changing, we define instantaneous angular velocity as the limit over arbitrarily short time intervals:
v = lim
∆tS0
∆u
∆t = du
dt 1instantaneous angular velocity2 (10.2) These definitions are analogous to those of average and instantaneous linear velocity in- troduced in Chapter 2. Just as we use the term speed for the magnitude of velocity, so we define angular speed as the magnitude of the angular velocity.
Velocity is a vector quantity, with magnitude and direction. Is angular velocity also a vector? Yes, but we’ll wait until the next chapter for the full vector description of rota- tional motion. In this chapter, it’s sufficient to know whether an object’s rotation is clock- wise (CW) or counterclockwise (CCW) about a fixed axis—as suggested by the curved arrow in Fig. 10.2. This restriction to a fixed axis is analogous to Chapter 2’s restriction to one-dimensional motion.
Angular and Linear Speed
Individual points on a rotating object undergo circular motion. Each point has an instan- taneous linear velocity v! whose magnitude is the linear speed v. We now relate this linear speed v to the angular speed v. The definition of angular measure in radians (Fig. 10.1) is u = s/r. Differentiating this expression with respect to time, we have
du dt = 1
r ds dt
because the radius r is constant. The left-hand side of this equation is the angular velocity v, as defined in Equation 10.2. Because s is the arc length—the actual distance traversed by a point on the rotating object—the term ds/dt is just the linear speed v, so v = v/r, or
v = vr (10.3)
Thus the linear speed of any point on a rotating object is proportional both to the angular speed of the object and to the distance from that point to the axis of rotation (Fig. 10.3).
✓TIp Radian Measure
Equation 10.3 was derived using the definition of angle in radians and therefore holds for only angular speed measured in radians per unit time. If you’re given other angular measures—degrees or revolutions, for example—you should convert to radians before using Equation 10.3.
s r The full circumference is 2pr, so 1 revolution is 2p radians. That makes 1 radian 360°>2p or about 57.3°.
Angle in radians is the ratio of arc s to radius r:
u = s>r. Here u is a little less than 1 radian.
r s u
u =
Figure 10.1 Radian measure of angles.
∆u The arm rotates through the angle ∆u in time ∆t, so its average angular velocity is v = ∆u>∆t.
Direction is
counterclockwise (CCW).
Figure 10.2 Average angular velocity.
Figure 10.3 Linear and rotational speeds.
vu vu v
Linear speed is proportional to distance from the rotation axis.
The point on the rim has the same angular speed v but a higher linear speed v than the inner point.
v = vr r
PheT: Ladybug Revolution
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170 Chapter 10 Rotational Motion
Angular Acceleration
If the angular velocity of a rotating object changes, then the object undergoes angular acceleration a, defined analogously to linear acceleration:
a = lim
∆tS0
∆v
∆t = dv
dt 1angular acceleration2 (10.4) Taking the limit gives the instantaneous angular acceleration; if we don’t take the limit, then we have an average over the time interval ∆t. The SI units of angular acceleration are rad/s2, although we sometimes use other units such as rpm/s or rev/s2.
Angular acceleration has the same direction as angular velocity—CW or CCW—if the angular speed is increasing, and the opposite direction if it’s decreasing. These situations are analogous to a car that’s speeding up (acceleration and velocity in the same direction) or braking (acceleration opposite velocity).
When a rotating object undergoes angular acceleration, points on the object speed up or slow down. Therefore, they have tangential acceleration dv/dt directed parallel or antiparallel to their linear velocity (Fig. 10.4). We introduced this idea of tangential accel- eration back in Chapter 3; here we can recast it in terms of the angular acceleration:
at = dv
dt = r dv
dt = ra 1tangential acceleration2 (10.5) Whether or not there’s angular acceleration, points on a rotating object also have radial accel- eration because they’re in circular motion. Radial acceleration is given, as usual, by ar = v2/r;
using v = vr from Equation 10.3, we can recast this equation in angular terms as ar = v2r.
Because angular velocity and acceleration are defined analogously to linear velocity and acceleration, all the relations among linear position, velocity, and acceleration au- tomatically apply among angular position, angular velocity, and angular acceleration. If
angular acceleration is constant, then all our constant-acceleration formulas of Chapter 2 apply when we make the substitutions u for x, v for v, and a for a. Table 10.1 summarizes
EvaluatE One revolution is 2p rad, and 1 min is 60 s, so we have v = 21 rpm = 121 rev/min212p rad/rev2
60 s/min = 2.2 rad/s The speed at the tip of a 28-m-long blade then follows from Equation 10.3: v = vr = 12.2 rad/s2128 m2 = 62 m/s.
assEss With v in radians per second, multiplying by length in me- ters gives correct velocity units of meters per second because radians
are dimensionless. ■
A wind turbine’s blades are 28 m long and rotate at 21 rpm. Find the angular speed of the blades in radians per second, and determine the linear speed at the tip of a blade.
IntErprEt This problem is about converting between two units of angular speed, revolutions per minute and radians per second, as well as finding linear speed given angular speed and radius.
DEvElop We’ll first convert the units to radians per second and then calculate the linear speed using Equation 10.3, v = vr.
Figure 10.4 Radial and tangential acceleration.
au
vu
at is the tangential component of acceleration a and is parallel to the linear velocity v.
r u at = ra
ar = v2r v
v
ar is the radial component, perpendicular to v.
u
u
u
Table 10.1 Angular and Linear Position, Velocity, and Acceleration
Linear Quantity Angular Quantity
Position x Angular position u
Velocity v = dx
dt Angular velocity v = du
dt Acceleration a = dv
dt = d2x
dt2 Angular acceleration a = dv dt = d2u
dt2
equations for Constant Linear Acceleration equations for Constant Angular Acceleration
v = 121v0 + v2 (2.8) v = 121v0 + v2 (10.6)
v = v0 + at (2.7) v = v0 + at (10.7)
x = x0 + v0 t + 12 at2 (2.10) u = u0 + v0 t + 12 at2 (10.8) v2 = v02+ 2a1x - x02 (2.11) v2= v02+ 2a1u - u02 (10.9) ExAmpLE 10.1 Angular Speed: A Wind Turbine
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