Angular Acceleration

LEARNING OBJECTIVES

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

• Describe uniform circular motion.
• Explain non-uniform circular motion.
• Calculate angular acceleration of an object.
• Observe the link between linear and angular acceleration.

Uniform Circular Motion and Gravitation discussed only uniform circular motion, which is motion in a circle at constant speed and, hence, constant angular velocity. Recall that angular velocity ω𝜔 was defined as the time rate of change of angle θ𝜃:

ω=ΔθΔt,𝜔=Δ𝜃Δ𝑡,

10.1

where θ𝜃 is the angle of rotation as seen in Figure 10.3. The relationship between angular velocity ω𝜔 and linear velocity v𝑣 was also defined in Rotation Angle and Angular Velocity as

v=rω𝑣=rω

10.2

or

ω=vr,𝜔=𝑣𝑟,

10.3

where r𝑟 is the radius of curvature, also seen in Figure 10.3. According to the sign convention, the counter clockwise direction is considered as positive direction and clockwise direction as negative

Figure 10.3 This figure shows uniform circular motion and some of its defined quantities.

Angular velocity is not constant when a skater pulls in her arms, when a child starts up a merry-go-round from rest, or when a computer’s hard disk slows to a halt when switched off. In all these cases, there is an angular acceleration, in which ω𝜔 changes. The faster the change occurs, the greater the angular acceleration. Angular acceleration α𝛼 is defined as the rate of change of angular velocity. In equation form, angular acceleration is expressed as follows:

α=ΔωΔt,𝛼=Δ𝜔Δ𝑡,

10.4

where ΔωΔ𝜔 is the change in angular velocity and ΔtΔ𝑡 is the change in time. The units of angular acceleration are (rad/s)/srad/s/s, or rad/s2rad/s2. If ω𝜔 increases, then α𝛼 is positive. If ω𝜔 decreases, then α𝛼 is negative.

EXAMPLE 10.1

Calculating the Angular Acceleration and Deceleration of a Bike Wheel

Suppose a teenager puts her bicycle on its back and starts the rear wheel spinning from rest to a final angular velocity of 250 rpm in 5.00 s. (a) Calculate the angular acceleration in rad/s2rad/s2. (b) If she now slams on the brakes, causing an angular acceleration of –87.3rad/s2–87.3rad/s2, how long does it take the wheel to stop?

Strategy for (a)

The angular acceleration can be found directly from its definition in α=ΔωΔt𝛼=Δ𝜔Δ𝑡 because the final angular velocity and time are given. We see that ΔωΔ𝜔 is 250 rpm and ΔtΔ𝑡 is 5.00 s.

Solution for (a)

Entering known information into the definition of angular acceleration, we get

α==ΔωΔt250 rpm5.00 s.𝛼=Δ𝜔Δ𝑡=250 rpm5.00 s.

10.5

Because ΔωΔ𝜔 is in revolutions per minute (rpm) and we want the standard units of rad/s2rad/s2 for angular acceleration, we need to convert ΔωΔ𝜔 from rpm to rad/s:

Δω==250revmin⋅2π radrev⋅1 min60 s26.2rads.Δ𝜔=250revmin⋅2π radrev⋅1 min60 s=26.2rads.

10.6

Entering this quantity into the expression for α𝛼, we get

α===ΔωΔt26.2 rad/s5.00 s5.24 rad/s2.𝛼=Δ𝜔Δ𝑡=26.2 rad/s5.00 s=5.24 rad/s2.

10.7

Strategy for (b)

In this part, we know the angular acceleration and the initial angular velocity. We can find the stoppage time by using the definition of angular acceleration and solving for ΔtΔ𝑡, yielding

Δt=Δωα.Δ𝑡=Δ𝜔𝛼.

10.8

Solution for (b)

Here the angular velocity decreases from 26.2 rad/s26.2 rad/s (250 rpm) to zero, so that ΔωΔ𝜔 is –26.2 rad/s–26.2 rad/s, and α𝛼 is given to be –87.3rad/s2–87.3rad/s2. Thus,

Δt==–26.2 rad/s–87.3rad/s20.300 s.Δ𝑡=–26.2 rad/s–87.3rad/s2=0.300 s.

10.9

Discussion

Note that the angular acceleration as the girl spins the wheel is small and positive; it takes 5 s to produce an appreciable angular velocity. When she hits the brake, the angular acceleration is large and negative. The angular velocity quickly goes to zero. In both cases, the relationships are analogous to what happens with linear motion. For example, there is a large deceleration when you crash into a brick wall—the velocity change is large in a short time interval.

If the bicycle in the preceding example had been on its wheels instead of upside-down, it would first have accelerated along the ground and then come to a stop. This connection between circular motion and linear motion needs to be explored. For example, it would be useful to know how linear and angular acceleration are related. In circular motion, linear acceleration is tangent to the circle at the point of interest, as seen in Figure 10.4. Thus, linear acceleration is called tangential acceleration at𝑎t.

Figure 10.4 In circular motion, linear acceleration a𝑎, occurs as the magnitude of the velocity changes: a𝑎 is tangent to the motion. In the context of circular motion, linear acceleration is also called tangential acceleration at𝑎t.

Linear or tangential acceleration refers to changes in the magnitude of velocity but not its direction. We know from Uniform Circular Motion and Gravitation that in circular motion centripetal acceleration, ac𝑎c, refers to changes in the direction of the velocity but not its magnitude. An object undergoing circular motion experiences centripetal acceleration, as seen in Figure 10.5. Thus, at𝑎t and ac𝑎c are perpendicular and independent of one another. Tangential acceleration at𝑎t is directly related to the angular acceleration α𝛼 and is linked to an increase or decrease in the velocity, but not its direction.

Figure 10.5 Centripetal acceleration ac𝑎c occurs as the direction of velocity changes; it is perpendicular to the circular motion. Centripetal and tangential acceleration are thus perpendicular to each other.

Now we can find the exact relationship between linear acceleration at𝑎t and angular acceleration α𝛼. Because linear acceleration is proportional to a change in the magnitude of the velocity, it is defined (as it was in One-Dimensional Kinematics) to be

at=ΔvΔt.𝑎t=Δ𝑣Δ𝑡.

10.10

For circular motion, note that v=rω𝑣=rω, so that

at=Δ(rω)Δt.𝑎t=ΔrωΔ𝑡.

10.11

The radius r𝑟 is constant for circular motion, and so Δ(rω)=r(Δω)Δ(rω)=𝑟(Δ𝜔). Thus,

at=rΔωΔt.𝑎t=𝑟Δ𝜔Δ𝑡.

10.12

By definition, α=ΔωΔt𝛼=Δ𝜔Δ𝑡. Thus,

at=rα,𝑎t=rα,

10.13

or

α=atr.𝛼=𝑎t𝑟.

10.14

These equations mean that linear acceleration and angular acceleration are directly proportional. The greater the angular acceleration is, the larger the linear (tangential) acceleration is, and vice versa. For example, the greater the angular acceleration of a car’s drive wheels, the greater the acceleration of the car. The radius also matters. For example, the smaller a wheel, the smaller its linear acceleration for a given angular acceleration α𝛼.

EXAMPLE 10.2

Calculating the Angular Acceleration of a Motorcycle Wheel

A powerful motorcycle can accelerate from 0 to 30.0 m/s (about 108 km/h) in 4.20 s. What is the angular acceleration of its 0.320-m-radius wheels? (See Figure 10.6.)

Figure 10.6 The linear acceleration of a motorcycle is accompanied by an angular acceleration of its wheels.

Strategy

We are given information about the linear velocities of the motorcycle. Thus, we can find its linear acceleration at𝑎t. Then, the expression α=atr𝛼=𝑎t𝑟 can be used to find the angular acceleration.

Solution

The linear acceleration is

at===ΔvΔt30.0 m/s4.20 s7.14m/s2.𝑎t=Δ𝑣Δ𝑡=30.0 m/s4.20 s=7.14m/s2.

10.15

We also know the radius of the wheels. Entering the values for at𝑎t and r𝑟 into α=atr𝛼=𝑎t𝑟, we get

α===atr7.14m/s20.320 m22.3rad/s2.𝛼=𝑎t𝑟=7.14m/s20.320 m=22.3rad/s2.

10.16

Discussion

Units of radians are dimensionless and appear in any relationship between angular and linear quantities.

So far, we have defined three rotational quantities— θ, ω𝜃, 𝜔, and α𝛼. These quantities are analogous to the translational quantities x, v𝑥, 𝑣, and a𝑎. Table 10.1 displays rotational quantities, the analogous translational quantities, and the relationships between them.

Rotational	Translational	Relationship
θ𝜃	x𝑥	θ=xr𝜃=𝑥𝑟
ω𝜔	v𝑣	ω=vr𝜔=𝑣𝑟
α𝛼	a𝑎	α=atr𝛼=𝑎𝑡𝑟

MAKING CONNECTIONS: TAKE-HOME EXPERIMENT

Sit down with your feet on the ground on a chair that rotates. Lift one of your legs such that it is unbent (straightened out). Using the other leg, begin to rotate yourself by pushing on the ground. Stop using your leg to push the ground but allow the chair to rotate. From the origin where you began, sketch the angle, angular velocity, and angular acceleration of your leg as a function of time in the form of three separate graphs. Estimate the magnitudes of these quantities.