Physics #19: The Kinematics of Circular Motion Adam Kirsch Andrea Hayes Larry Ottman Brenda Meery Bradley Hughes Mara Landers Art Fortgang Lori Jordan CK12 Editor James H Dann, Ph.D. Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) www.ck12.org AUTHORS Adam Kirsch To access a customizable version of this book, as well as other Andrea Hayes interactive content, visit www.ck12.org Larry Ottman Brenda Meery Bradley Hughes Mara Landers Art Fortgang CK-12 Foundation is a non-profit organization with a mission to Lori Jordan reduce the cost of textbook materials for the K-12 market both CK12 Editor in the U.S. and worldwide. Using an open-content, web-based James H Dann, Ph.D. collaborative model termed the FlexBook®, CK-12 intends to pioneer the generation and distribution of high-quality educational CONTRIBUTORS content that will serve both as core text as well as provide an Chris Addiego adaptive environment for learning, powered through the FlexBook Ari Holtzman Platform®. Antonio De Jesus López Tinyen Shih Copyright © 2012 CK-12 Foundation, www.ck12.org The names “CK-12” and “CK12” and associated logos and the terms “FlexBook®” and “FlexBook Platform®” (collectively “CK-12 Marks”) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link http://www.ck12.org/saythanks (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution/Non- Commercial/Share Alike 3.0 Unported (CC BY-NC-SA) License (http://creativecommons.org/licenses/by-nc-sa/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated herein by this reference. Complete terms can be found at http://www.ck12.org/terms. Printed: February 3, 2013 iii Contents www.ck12.org Contents 1 Radian Measure – An Introductory Video1 2 Radian Measure – A Deeper Look2 3 Linear vs. Angular Velocity and How They Relate 11 4 Linear vs. Angular Velocity – A Deeper Look 12 5 Angular Velocity – Some Final Points 19 6 Kinematics of Rotation – The Big 4 When Things Do Circles 22 iv www.ck12.org Chapter 1. Radian Measure – An Introductory Video CHAPTER 1 Radian Measure – An Introductory Video Here you’ll learn what radian measure is, and how to find the radian values for common angles on the unit circle. While working on an experiment in your school science lab, your teacher asks you to turn up a detector by rotating p the knob 2 radians. You are immediately puzzled, since you don’t know what a radian measure is or how far to turn the knob. Read this Concept, and at its conclusion, you will be able to turn the knob by the amount your teacher requested. Watch This MEDIA Click image to the left for more content. James Sousa:Radian Measure 1 www.ck12.org CHAPTER 2 Radian Measure – A Deeper Look Learning Objectives A student will be able to: • Define radian measure. • Convert angle measure from degrees to radians and from radians to degrees. Introduction In this lesson you will be introduced to the radian as a common unit of angle measure. It is important to become proficient converting back and forth between degrees and radians. Eventually, much like learning a foreign language, you will become comfortable with radian measure when you can learn to “think” in radians instead of always converting from degree measure. It is essential that you master this. When degrees are not part of a trigonometric function and are used in an equation (as with the equations for rotational kinematics – the subject for today) angles MUST be expressed in radians. The reason for this has to do with what a radian is. As you will see, the degree measure is a completely arbitrary one that we decided upon long ago, whereas the radian is intimately connected to the fundamental definition of a circle and therefore can be used directly in computations. Understanding Radian Measure Many units of measure come from seemingly arbitrary and archaic roots. Some even change over time. The meter, for example was originally intended to be based on the circumference of the earth and now has an amazingly complicated scientific definition! We typically use degrees to measure angles. Exactly what is a degree? A degree is 1=360th of a complete rotation around a circle. Radians are alternate units used to measure angles in trigonometry. Just as it sounds, a radian is based on the radius of a circle. One radian is the angle created by bending the radius length around the arc of a circle. Because a radian is based on an actual part of the circle rather than an arbitrary division, it is a much more natural unit of angle measure for upper level mathematics and will be especially useful in this unit in physics. 2 www.ck12.org Chapter 2. Radian Measure – A Deeper Look What if we were to rotate all the way around the circle? Continuing to add radius lengths, we find that it takes a little more than 6 of them to complete the rotation. But the arc length of a complete rotation is really the circumference! The circumference is equal to 2p times the length of the radius. 2p is approximately 6:28, so the circumference is a little more than 6 radius lengths. Or, in terms of radian measure, a complete rotation (360 degrees) is 2p radians. 360 degrees = 2p radians With this as our starting point, we can find the radian measure of other angles easily. Half of a rotation, or 180 degrees, must therefore be p radians, and 90 degrees must be one-half pi. Complete the table below: TABLE 2.1: Angle in Degrees Angle in Radians p 90 2 45 30 60 75 3 www.ck12.org Because 45 is half of 90, half of one-half p is one-fourth p. 30 is one-third of a right angle, so multiplying gives: p 1 p × = 2 3 6 and because 60 is twice as large as 30: p 2p p 2 × = = 6 6 3 Here is the completed table: TABLE 2.2: Angle in Degrees Angle in Radians p 90 2 p 45 4 p 30 6 p 60 3 5p 75 12 The last value was found by adding the radian measures of 30 and 45: p p 3 × p 2 × p + = + 4 6 3 × 4 2 × 6 3p 2p 5p + = 12 12 12 There is a formula to help you convert between radians and degrees that you may already have discovered, however, most angles that you will commonly use to describe what takes place int he real world can not be found easily from the values in this table. Instead, it is usually most useful to consider conversions when going back and forth between radians and degrees. The key is remembering this one conversion: 360 degrees = 2p radians When going from 1.35 radians to degrees: 1.35 rad (360 degrees/2p rad) = 77.3 degrees If wanting 192 degrees expressed in radians: 192 degrees (2p rad/360 degrees) = 3.35 radians As always when converting units, let the units and how they must cancel be your guide when determining which of these conversion factors you need to use. That way, you will only have to remember one thing – how many radians and degrees are in a full circle! One other note – radians are actually dimensionless. You see here either "rad" or the word "radians" used as units for angular quantities measured in radians just as reminders. But radians actually are dimensionless. Here is why. Refer again to the first diagram above. The arc length (s), the angle associated with this arc(q), and the radius ( r) are all related by the following relationship: q = sr When s = one radius (r), then q = r/r, the r’s cancel as do their units of meters, and we are left with the value of 1. But one what? Well, we call this one radian, and therefore we have the definition of what a radian is. However in terms of units, it really has none! 4 www.ck12.org Chapter 2. Radian Measure – A Deeper Look Radians, Degrees, and a Calculator Most scientific and graphing calculators have a [MODE] setting that will allow you to either convert between the two, or to find approximations for trig functions using either measure. It is important that if you are using your calculator to estimate a trig function that you know which mode you are using. Look at the following screen: If you entered this expecting to find the sine of 30 degrees you would realize based on the last chapter that something 1 is wrong because it should be 2 . In fact, as you may have suspected, the calculator is interpreting this as 30radians. In this case, changing the mode to degrees and recalculating we give the expected result. Scientific calculators will usually have a 3−letter display that shows either DEG or RAD to tell you which mode you are in. Always check before calculating a trig ratio!! In general in this class you will want your calculator set to degrees. Yes degrees! You are probably wondering why degrees, given that we just got done discussing how most of the numbers in this unit must be in radians.