Slide 1 / 132 Slide 2 / 132 Rotational Motion www.njctl.org Slide 3 / 132 Slide 4 / 132 Table of Contents How to Use this File Click on the topic to go to that section Each topic is composed of brief direct instruction · Rotational Kinematics Review · There are formative assessment questions after every topic · · Rotational Dynamics denoted by black text and a number in the upper left. · Rotational Kinetic Energy >Students work in groups to solve these problems but use · Angular Momentum student responders to enter their own answers. >Designed for SMART Response PE student response systems. >Use only as many questions as necessary for a sufficient number of students to learn a topic. Full information on how to teach with NJCTL courses can be · found at njctl.org/courses/teaching methods Slide 5 / 132 Slide 6 / 132 What is Rotational Motion? Linear Kinematics and Dynamics dealt with the motion of "point Rotational Kinematics particles" moving in straight lines. Life is more complex than that. Review Objects have an underlying structure, so it cannot be assumed that a force applied anywhere on the object results in the same kind of motion. In football (American and the rest of the world), if a player is struck from the front, at the ankle, he will fall forward, pivoting about his foot. The same force applied in his midsection will push him backwards. Return to Table The player is rotating about his ankle in the first case, and of Contents translating (linear) backwards in the second case.
Slide 7 / 132 Slide 8 / 132 Axis of Rotation Axis of Rotation In the first case, the player was rotated about his axis of No - if you were to spin this donut around its center, the axis of rotation - a line parallel to the ground and through his ankle, rotation would be in the donut hole, pointing out of the page. perpendicular to the force. Assume that all the objects that we have been dealing with are rigid bodies, that is, they keep their shape and are not deformed in any way by their motion. Here's a sphere rotating about its axis of rotation - the vertical red line. Does the axis of rotation have to be part of the rigid body? "Chocolate dip,2011-11-28" by Pbj2199 - Own work. Licensed under Creative Commons Attribution-Share Alike 3.0 via Wikimedia "Rotating Sphere". Licensed under Creative Commons Attribution-Share Alike 3.0 via Wikimedia Commons - http://commons.wikimedia.org/wiki/ Commons - http://commons.wikimedia.org/wiki/File:Chocolate_dip, File:Rotating_Sphere.gif#mediaviewer/File:Rotating_Sphere.gif 2011-11-28.jpg#mediaviewer/File:Chocolate_dip,2011-11-28.jpg Slide 9 / 132 Slide 10 / 132 Rotational Equivalents to Angular Displacement Linear (Translational) Motion There are several theories why a circle has 360 degrees. Linear motion is described by displacement, velocity and Here's a few - if you're interested, there's a lot of information acceleration. that can be found on the web. Rotational motion is described by angular displacement, Some people believe it stems from the ancient Babylonians angular velocity and angular acceleration. who had a number system based on 60 instead of our base 10 system. It is also important to define the axis of rotation - objects will have different properties depending on this choice. Once the Others track it back to the Persian or biblical Hebrew calendars choice is made, it must remain the same to describe the of 12 months of 30 days each. motion. If the axis is changed, then a different rotational motion occurs. But - it has nothing to do with the actual geometry of the circle. There is a more natural unit - the radian, which is the rotational equivalent of displacement (Δs). Slide 11 / 132 Slide 12 / 132 Angular Displacement Radian Angular displacement is unit less since it is the ratio of two As a rigid disc (circle) rotates about its center, the angle of distances. But, we will say that angular displacement is rotation, θ, is defined as the ratio of the subtended arc length, s, measured in radians (abbreviated "rad") . A point moving around to the radius of the circle, and is named angular displacement . the entire circumference will travel 2πr as an angle of 360 0 is swept through. B Using the angular displacement definition: A B Rotational angles are now defined A in geometric terms, not based on an old calendar or arbitrary numbering system.
Slide 13 / 132 Slide 14 / 132 1 What is the angular displacement for an arc length (s) that is Angular Displacement equal to the radius of the circular rigid body? Something interesting - look B 3 at the three concentric A 0.5 rad B 2 circles drawn on the rigid disc the radii, arc lengths s r B 1 B 1.0 rad and the points A r and B r . C 0.5π rad As the disk rotates, each A 1 A 2 A 3 point A moves to point B, covering the SAME angle θ, D 1.0π rad but covering a different arc length s r . E 1.5π rad Thus, all points on a rotating rigid disc move through the same angle, θ, but different arc lengths s, and different radii, r. Slide 15 / 132 Slide 16 / 132 2 A record spins 4 times around its center. It makes 4 revolutions. 3 A record spins 4 times around its center. It makes 4 How many radians did it pass? revolutions. How many radians did it pass? A π rad A π rad B 2π rad B 2π rad C 4π rad C 4π rad D 8π rad D 6π rad E 8π rad Slide 17 / 132 Slide 18 / 132 4 A circular hoop of radius 0.86 m rotates π/3 rad about its 5 There are two people on a merry go round. Person A is 2.3 m from center. A small bug is on the hoop - what distance does it the axis of rotation. Person B is 3.4 m from the axis of rotation. The merry go round moves through an angular displacement of travel (arc length) during this rotation? π/4. What linear displacement (arc length) is covered by both people? Compare and contrast these motions. A 0.30 m B 0.90 m C 1.4 m D 2.7 m E 3.2 m
Slide 19 / 132 Slide 20 / 132 Other Angular Quantities Other Angular Quantities Angular displacement has now been be related to Linear Define average Angular Velocity (ω avg ) as the change in displacement. angular displacement over time, and take the limit as Δt approaches zero for the instantaneous angular velocity (ω). Next, let's relate angular velocity and angular acceleration to their linear equivalents: Similarly for average Angular Acceleration (α) and instantaneous angular acceleration (α). Slide 21 / 132 Slide 22 / 132 Other Angular Quantities Angular and Linear Velocity Start with angular velocity, and substitute in the linear Since ω and α are both related to θ, and θ is the same for all displacement for the angular displacement. points on a rotating rigid body, ω and α are also the same for all points on a rotating rigid body. But the same is not true for linear velocity and acceleration. r is constant, so move it outside the derivative Let's relate the linear velocity and acceleration to their angular equivalents. This confirms what you feel on a merry go round - the further away from the center you move, the faster you feel you're going - you're feeling the linear velocity! Slide 23 / 132 Slide 24 / 132 Angular and Linear Acceleration Angular and Linear Quantities Summary Start with angular acceleration, and substitute in the linear velocity for the angular velocity. Angular Linear Relationship r is constant, so move Displacement it outside the derivative Velocity Acceleration
Slide 25 / 132 Slide 26 / 132 Angular Velocity sign Angular Velocity sign We're familiar with how to assign positive and negative values to Not really. Not at all. displacement. Typically if you move to the right or up, we give that a positive displacement (of course, this is arbitrary, but it's a pretty From your math classes, you know that the horizontal axis through a standard convention). circle is labeled as 0 0 , and angles are measured in a counter clockwise direction. B B How do we assign a "sign" Once we agree on that, let's for angular displacement, look at the definition of velocity or acceleration? angular velocity: A A Does "right" or "up" work with rotational motion? Does it work with a circle? Slide 27 / 132 Slide 28 / 132 6 Explain why a disc that rotates clockwise leads to a negative value Angular Velocity sign for angular velocity, ω. You can use an example to show your point. Students type their answers here As the disc rotates in a counter clockwise fashion, θ increases, so Δθ is positive. Since dt is also positive, dθ/dt must be positive. Thus, Counter clockwise rotations result in a positive ω. B A Slide 29 / 132 Slide 30 / 132 Angular Acceleration sign Angular Acceleration sign Start with the definition of angular acceleration: Here they are: There are four cases here, all similar to the case of linear ω increases in counter clockwise direction - α is positive acceleration. Can you figure them out? ω decreases in the counter clockwise direction - α is negative Here's a hint - for linear acceleration, if an object's velocity is ω increases in clockwise direction - α is negative increasing in the same direction of its displacement, the acceleration ω decreases in the clockwise direction - α is positive is positive. These conventions are arbitrary, and in this case are If the object's velocity is decreasing in the direction of its based on the axis of rotation, but once chosen, they displacement, then its acceleration is negative. need to stay consistent for solving problems. There will be problems where the choice is not so obvious, but just make a choice and stay consistent.
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