Chapter 10: Rotational Kinematics and Energy
Chapter Review


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Chapter Review


In this, and the next chapter we study rotational kinematics and dynamics. Rotational motion is every bit as important as the linear (or translational) motion that we've been studying thus far. As you go through these chapters try to notice how the study of rotational motion parallels that of translational motion. Many of the concepts are the same, as well as the mathematical treatment, we only need to alter the physical interpretation of these concepts from that of a translating body to a rotating one.

10-1 Angular Position, Velocity, and Acceleration

Just as with linear motion we describe rotational motion with three basic quantities analogous to displacement, velocity, and acceleration. We shall call these quantities angular position, angular velocity, and angular acceleration.

When we describe the rotation of an object we think of the object as rotating about an axis. If you consider a line drawn from the axis of rotation to the object, the angular position of the object, denoted by q, is the angle between that line and an arbitrarily chosen reference line. The reference line plays the same role as the arbitrarily chosen origin of a coordinate system. In SI the unit of angular measure, for rotational motion, is the radian. Angular measure is actually dimensionless, but radian measure works better in describing motion than does degrees. Angular position plays the same role for rotational motion that position does for translational motion. In keeping with the usual convention, if q is measured counterclockwise it is taken to be a positive angle and if it is measured clockwise take it to be negative.

To describe how rapidly an object rotates about an axis we use the quantity angular velocity w. This quantity plays the same role for rotational motion that velocity does for linear motion. Angular velocity is the rate of change of angular position. Often it is useful to use the average rate of change, or average angular velocity

.

The SI unit of angular velocity is rad/s = s-1 (since rad is dimensionless). It is also sometimes useful to use the instantaneous angular velocity

.

To maintain consistency with the sign convention for angular displacement Dq, angular velocity is negative if the object rotates clockwise and positive if it rotates counterclockwise.

To describe how the angular velocity of an object changes we use the quantity angular acceleration a. This quantity plays the same role for rotational motion that acceleration (a) does for linear motion. Angular acceleration is the rate of change of angular velocity. Often it is useful to use the average rate of change, or average angular acceleration

.

The SI unit of angular acceleration is rad/s2 = s-2 (since rad is dimensionless). It is also sometimes useful to use the instantaneous angular acceleration

.

The sign of a depends on the sign of Dw. If Dw is negative then a is negative; if Dw is positive then a is positive.


Example 10.1 Turn on the fan: On a hot day you turn on a small fan to help you cool off. The tip on one of the blades of the fan might go through 1.25 revolutions at an average angular velocity of 7.85 rad/s. (a) What is the angular displacement of the tip of the blade, and (b) how much time does it take to go through that displacement?

Picture the Problem The picture shows the blades of a fan with one tip indicated by the black dot. In this diagram the fan is rotating clockwise.

Strategy For part (a) we must translate between the number of revolutions and the number of radians to get the SI angular displacement. For part (b) we'll make use of the definition of average angular velocity.

Solution
Part (a)
1. Convert from revolutions to radians:

Part (b)
1. Use definition of wav to solve for Dt:
2. Calculate the numerical result:

Insight Notice the minus signs, this is done only by convention because the blade rotates clockwise.



Physlet Illustration: Angular Velocity

Interactive Help
on      off
A wheel turns at a constant rate. Determine its angular velocity, omega, in radians/sec. Start

Hints:

  1. Determine how long it takes the wheel to make one complete revolution.
  2. How many radians does it turn in that single revolution?



Practice Quiz

 
Which quantity is used to describe how rapidly an object rotates?
angular postition
angular velocity
angular acceleration
[none of the above]
 
If an object is seen to rotate faster and faster, which quantity best describes this aspect of its motion?
angular position
angular velocity
angular acceleration
[none of the above]

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your answer: angular velocity

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your answer: angular acceleration

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10-2 Rotational Kinematics

One thing you may have noticed about the three quantities described in the previous section is that the mathematical relationships between them is exactly the same as the relationships between the corresponding quantities for translational motion. This means that when looking for a way to describe rotational motion, we can follow the prescription already laid out for translational motion. All of the equations used in chapter two for describing motion with constant velocity and constant acceleration also apply to rotational motion. The only mathematical difference is that the names of the variables are changed in the following way

.

Of course, along with the change of variables, you should also change the physical picture in your mind of the type of motion you are describing. Nevertheless, the mathematical descriptions are identical.

So, to describe rotational motion, we make the above replacements and reuse the equations in chapter two for one-dimensional kinematics.

(A) Motion with Constant Angular Velocity

(B) Motion with Constant Angular Acceleration


Example 10.2 Turn on the fan II: Suppose that the blades of the fan discussed in example 10.1 rotate with a constant angular acceleration for the first 1.00 seconds. (a) What is the instantaneous angular velocity of the blade at t = 1.00 seconds after being turned on, and (b) what is the angular acceleration?

Picture the Problem The picture shows the blades of a fan with one tip indicated by the black dot. This time the fan is rotating counterclockwise.

Strategy In both parts we make use of the expressions for uniformly accelerated motion to obtain a solution. Let's make use of our freedom to define qo = 0 and to = 0. Also notice that since the fan is just being turned on wo = 0.

Solution
Part (a)
1. Choose an appropriate expression for finding w based on the given information:
2. Solve for angular velocity at the appropriate time:

Part (b)
1. Choose an appropriate expression for finding a based on the given information:
2. Use the know data to get the numerical result:

Insights In this example all of the values are positive: for q and w because it rotates counterclockwise and for a because the angular speed increases counterclockwise. Try to see how this problem could have been solved for translational motion way back in chapter two. If you think about it, you actually have already done this stuff!



Physlet Illustration: Angular Acceleration

Interactive Help
on      off
Starting from rest, a wheel turns with constant angular acceleration. Its instantaneous angular velocity, omega, is shown in radians/sec. Determine the angular acceleration, alpha, in radians/s2. Start

Hints:

  1. Choose a specific time interval, and determine the angular velocity at the beginning and end of that interval.
  2. Divide the change in angular velocity by the time it took to make that change.




Example 10.3 Turn off the fan: Now suppose that the blades of our favorite fan has a top rotation rate of 25 revolutions per second. Once we've cooled down enough we turn off the power and it takes 6.65 seconds for the blades to come to rest with a constant angular acceleration. (a) What is the angular acceleration of the blades, and (b) through what angular displacement do the blades turn while coming to rest?

Solution: The problem doesn't give us a direction of rotation so we are free to choose. Let's choose counterclockwise as a default.

Given: w0 = 25 rev/s, w = 0, t = 6.65 s. Find: (a) a, (b) q

First I'll convert the initial angular velocity to SI units

.

Now, for part (a), I can determine a by using

.

Calculating the answer to part

.

Insight Notice that because of the original rotation rate I only give answers to two significant digits.


Practice Quiz

 
In comparing linear and rotational motion, angular position is most similar to which linear variable?
x
v
a
t
[none of the above]
 
How long would it take an object rotating at constant speed of 17.0 rad/s to rotate through 235 radians?
0.0723 s
3.4 N
5.0 N
13.8 s
3.1 N
 
If an object rotates at 2.50 rpm, what is the equivilent angular speed in rad/s?
2.50
15.7
0.262
0.0417
25.0

your answer: x

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your answer: 13.8 s

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your answer: 0.262

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