Rotation Angle and Angular Velocity

LEARNING OBJECTIVES

By the end of this section, you will be able to:

• Define arc length, rotation angle, radius of curvature and angular velocity.
• Calculate the angular velocity of a car wheel spin.

In Kinematics, we studied motion along a straight line and introduced such concepts as displacement, velocity, and acceleration. Two-Dimensional Kinematics dealt with motion in two dimensions. Projectile motion is a special case of two-dimensional kinematics in which the object is projected into the air, while being subject to the gravitational force, and lands a distance away. In this chapter, we consider situations where the object does not land but moves in a curve. We begin the study of uniform circular motion by defining two angular quantities needed to describe rotational motion.

Rotation Angle

When objects rotate about some axis—for example, when the CD (compact disc) in Figure 6.2 rotates about its center—each point in the object follows a circular arc. Consider a line from the center of the CD to its edge. Each pit used to record sound along this line moves through the same angle in the same amount of time. The rotation angle is the amount of rotation and is analogous to linear distance. We define the rotation angle ΔθΔ𝜃 to be the ratio of the arc length to the radius of curvature:

Δθ=Δsr.Δ𝜃=Δ𝑠𝑟.

6.1

Figure 6.2 All points on a CD travel in circular arcs. The pits along a line from the center to the edge all move through the same angle ΔθΔ𝜃 in a time ΔtΔ𝑡.

Figure 6.3 The radius of a circle is rotated through an angle ΔθΔ𝜃. The arc length ΔsΔs is described on the circumference.

The arc lengthΔsΔ𝑠 is the distance traveled along a circular path as shown in Figure 6.3 Note that r𝑟 is the radius of curvature of the circular path.

We know that for one complete revolution, the arc length is the circumference of a circle of radius r𝑟. The circumference of a circle is 2πr2π𝑟. Thus for one complete revolution the rotation angle is

Δθ=2πrr=2π.Δ𝜃=2π𝑟𝑟=2π.

6.2

This result is the basis for defining the units used to measure rotation angles, ΔθΔ𝜃 to be radians (rad), defined so that

2πrad=1 revolution.2πrad=1 revolution.

6.3

A comparison of some useful angles expressed in both degrees and radians is shown in Table 6.1.

Degree Measures	Radian Measure
30º30º	π6𝜋6
60º60º	π3𝜋3
90º90º	π2𝜋2
120º120º	2π32π3
135º135º	3π43π4
180º180º	π𝜋

Figure 6.4 Points 1 and 2 rotate through the same angle (ΔθΔ𝜃), but point 2 moves through a greater arc length (Δs)Δ𝑠 because it is at a greater distance from the center of rotation (r)(𝑟).

If Δθ=2πΔ𝜃=2𝜋 rad, then the CD has made one complete revolution, and every point on the CD is back at its original position. Because there are 360º360º in a circle or one revolution, the relationship between radians and degrees is thus

2πrad=360º2𝜋rad=360º

6.4

so that

1rad=360º2π≈57.3º.1rad=360º2π≈57.3º.

6.5

Angular Velocity

How fast is an object rotating? We define angular velocity ω𝜔 as the rate of change of an angle. In symbols, this is

ω=ΔθΔt,𝜔=Δ𝜃Δ𝑡,

6.6

where an angular rotation ΔθΔ𝜃 takes place in a time ΔtΔ𝑡. The greater the rotation angle in a given amount of time, the greater the angular velocity. The units for angular velocity are radians per second (rad/s).

Angular velocity ω𝜔 is analogous to linear velocity v𝑣. To get the precise relationship between angular and linear velocity, we again consider a pit on the rotating CD. This pit moves an arc length ΔsΔ𝑠 in a time ΔtΔ𝑡, and so it has a linear velocity

v=ΔsΔt.𝑣=Δ𝑠Δ𝑡.

6.7

From Δθ=ΔsrΔ𝜃=Δ𝑠𝑟 we see that Δs=rΔθΔ𝑠=𝑟Δ𝜃. Substituting this into the expression for v𝑣 gives

v=rΔθΔt=rω.𝑣=𝑟Δ𝜃Δ𝑡=rω.

6.8

We write this relationship in two different ways and gain two different insights:

v=rω or ω=vr.𝑣=rω or 𝜔=𝑣𝑟.

6.9

The first relationship in v=rω or ω=vr𝑣=rω or 𝜔=𝑣𝑟 states that the linear velocity v𝑣 is proportional to the distance from the center of rotation, thus, it is largest for a point on the rim (largest r𝑟), as you might expect. We can also call this linear speed v𝑣 of a point on the rim the tangential speed. The second relationship in v=rω or ω=vr𝑣=rω or 𝜔=𝑣𝑟 can be illustrated by considering the tire of a moving car. Note that the speed of a point on the rim of the tire is the same as the speed v𝑣 of the car. See Figure 6.5. So the faster the car moves, the faster the tire spins—large v𝑣 means a large ω𝜔, because v=rω𝑣=rω. Similarly, a larger-radius tire rotating at the same angular velocity (ω𝜔) will produce a greater linear speed (v𝑣) for the car.

Figure 6.5 A car moving at a velocity v𝑣 to the right has a tire rotating with an angular velocity ω𝜔.The speed of the tread of the tire relative to the axle is v𝑣, the same as if the car were jacked up. Thus the car moves forward at linear velocity v=rω𝑣=rω, where r𝑟 is the tire radius. A larger angular velocity for the tire means a greater velocity for the car.

EXAMPLE 6.1

How Fast Does a Car Tire Spin?

Calculate the angular velocity of a 0.300 m radius car tire when the car travels at 15.0m/s15.0m/s (about 54km/h54km/h). See Figure 6.5.

Strategy

Because the linear speed of the tire rim is the same as the speed of the car, we have v=15.0 m/s.𝑣=15.0 m/s. The radius of the tire is given to be r=0.300 m.𝑟=0.300 m. Knowing v𝑣 and r𝑟, we can use the second relationship in v=rω, ω=vr𝑣=rω, 𝜔=𝑣𝑟 to calculate the angular velocity.

Solution

To calculate the angular velocity, we will use the following relationship:

ω=vr.𝜔=𝑣𝑟.

6.10

Substituting the knowns,

ω=15.0m/s0.300m=50.0rad/s.𝜔=15.0m/s0.300m=50.0rad/s.

6.11

Discussion

When we cancel units in the above calculation, we get 50.0/s. But the angular velocity must have units of rad/s. Because radians are actually unitless (radians are defined as a ratio of distance), we can simply insert them into the answer for the angular velocity. Also note that if an earth mover with much larger tires, say 1.20 m in radius, were moving at the same speed of 15.0 m/s, its tires would rotate more slowly. They would have an angular velocity

ω=(15.0m/s)/(1.20m)=12.5rad/s.𝜔=(15.0m/s)/(1.20m)=12.5rad/s.

6.12

Both ω𝜔 and v𝑣 have directions (hence they are angular and linear velocities, respectively). Angular velocity has only two directions with respect to the axis of rotation—it is either clockwise or counterclockwise. Linear velocity is tangent to the path, as illustrated in Figure 6.6.

TAKE-HOME EXPERIMENT

Tie an object to the end of a string and swing it around in a horizontal circle above your head (swing at your wrist). Maintain uniform speed as the object swings and measure the angular velocity of the motion. What is the approximate speed of the object? Identify a point close to your hand and take appropriate measurements to calculate the linear speed at this point. Identify other circular motions and measure their angular velocities.

Figure 6.6 As an object moves in a circle, here a fly on the edge of an old-fashioned vinyl record, its instantaneous velocity is always tangent to the circle. The direction of the angular velocity is clockwise in this case.