Rotational Motion used for any commercial purpose without the - PDF document

Slide 1 / 138 Slide 2 / 138 New Jersey Center for Teaching and Learning Progressive Science Initiative This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be Rotational Motion used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. Click to go to website: www.njctl.org www.njctl.org Slide 3 / 138 Slide 4 / 138 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 · Axis of Rotation and Angular Properties · There are formative assessment questions after every topic · · Rotational Kinematics denoted by black text and a number in the upper left. · Rotational Dynamics >Students work in groups to solve these problems but use · Rotational Kinetic Energy student responders to enter their own answers. · Angular Momentum >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 / 138 Slide 6 / 138 What is Rotational Motion? Up until now, we have treated everything as if it were a either a Axis of Rotation and point or a shape, but we would find its center of mass, and then pretend it acted like a point at its center of mass. Angular Properties Continuing with our thread of starting simple and then layering on more reality, it is now time to address the fact that things also rotate - and many interesting applications arise from this. What are some types of rotating objects and how are they important to society? Return to Table of Contents

Slide 7 / 138 Slide 8 / 138 Axis of Rotation What is Rotational Motion? The list is endless. They are all rotating about a line somewhere within the object called the axis of rotation. Here's some examples: We're also going to assume that all these objects are rigid Turbine generators that transfer mechanical energy into bodies, that is, they keep their shape and are not deformed in · electrical energy and supply electricity to businesses and any way by their motion. homes. Helicopter blades that lift the helicopter into the air. · Here's a sphere rotating about Electric motors that power cars, circular saws and vacuums · its axis of rotation - the vertical cleaners. red line. Swivel chairs. · Curve balls thrown by baseball pitchers. · Does the axis of rotation have Automobile and truck tires that propel them down the road. · to be part of the rigid body? What is a common feature of these examples? "Rotating Sphere". Licensed under Creative Commons Attribution-Share Alike 3.0 via Wikimedia Commons - http://commons.wikimedia.org/wiki/ File:Rotating_Sphere.gif#mediaviewer/File:Rotating_Sphere.gif Slide 9 / 138 Slide 10 / 138 Angular Displacement Axis of Rotation No - if you were to spin this donut around its center, the axis of There are several theories why a circle has 360 degrees. rotation would be in the donut hole, pointing out of the pagel Here's a few - if you're interested, there's a lot of information that can be found on the web. Some people believe it stems from the ancient Babylonians who had a number system based on 60 instead of our base 10 system. Others track it back to the Persian or biblical Hebrew calendars of 12 months of 30 days each. But - it has nothing to do with the actual geometry of the circle. There is a more natural unit - the radian. "Chocolate dip,2011-11-28" by Pbj2199 - Own work. Licensed under Creative Commons Attribution-Share Alike 3.0 via Wikimedia Commons - http://commons.wikimedia.org/wiki/File:Chocolate_dip, 2011-11-28.jpg#mediaviewer/File:Chocolate_dip,2011-11-28.jpg Slide 11 / 138 Slide 12 / 138 Angular Displacement Angular Displacement Let's look at this circle and assume it's rotating about its middle The point traveled a distance of s - so the axis of rotation is pointing out of the board. B along the circumference, and swept out an angle θ. We can also say that the angle θ B "subtends" an arc length of s. A Start with a piece of the circle at Note that the points A and B are point A. As the circle rotates always at the same distance, r, A counterclockwise, the piece of from the axis of rotation. That's the circle reaches point B. what it means to be a circle!

Slide 13 / 138 Slide 14 / 138 Angular Displacement Radian We will now define the angle of rotation, θ, as the ratio of the Angular displacement is unitless since it is the ratio of two arc length, s, to the radius of the circle. Can you see how this distances. But, we will say that angular displacement is is a much more natural definition of the angle of rotation then measured in radians . Let's relate this to concepts that we're basing it on an old calendar or arbitrary numbering system? pretty familiar with. We know degrees, and we know that when a point on a circle B rotates and comes back to the same point, it has performed one revolution - we start at point A, and rotate until we come back to point A. We will call this angle A B of rotation, θ, the angular displacement . What distance, s, was covered? A Rotational angles are now defined How many degrees were swept in geometric terms - as the ratio of by this full rotation? an arc length and the radius of the circle. Slide 15 / 138 Slide 16 / 138 Radian Radian The point moved around the entire circumference, so it traveled When an object makes one complete revolution, it sweeps out an 2πr while an angle of 360 0 was swept through. Using the angular angle of 360 0 or 2π radians. displacement definition: 1 radian = 57.3 0 The radian is frequently abbreviated as rad. The only thing you have to be careful about now is the settings on your calculator. Up until now, you probably just had angles set for degrees. B You now need to set your calculator for radians. Please ask a classmate or your teacher for help on this. A Slide 17 / 138 Slide 17 (Answer) / 138 1 What is the angle inside a circle, in radians, that subtends an arc 1 What is the angle inside a circle, in radians, that subtends an arc length of 0.25 m? The radius of the circle is 5.0 m. length of 0.25 m? The radius of the circle is 5.0 m. A 0.05 rad A 0.05 rad B 0.50 rad B 0.50 rad C 2 rad C 2 rad Answer A D 20 rad D 20 rad [This object is a pull tab]

Slide 18 / 138 Slide 18 (Answer) / 138 2 What is the value of π/2 radians in degrees? 2 What is the value of π/2 radians in degrees? A 0 0 A 0 0 B 45 0 B 45 0 C 90 0 C 90 0 Answer D 180 0 D 180 0 C [This object is a pull tab] Slide 19 / 138 Slide 19 (Answer) / 138 3 What is the angular displacement for an arc length (s) that is equal 3 What is the angular displacement for an arc length (s) that is equal to the radius of the circular rigid body? to the radius of the circular rigid body? A 0.5 rad A 0.5 rad B 1.0 rad B 1.0 rad Answer B C 0.5 π rad C 0.5 π rad D 1.0 π rad D 1.0 π rad [This object is a pull tab] Slide 20 / 138 Slide 20 (Answer) / 138 4 A record spins 4 times around its center. It makes 4 revolutions. 4 A record spins 4 times around its center. It makes 4 revolutions. How many radians did it pass? How many radians did it pass? A π rad A π rad B 2π rad B 2π rad C 4π rad C 4π rad Answer D D 8π rad D 8π rad [This object is a pull tab]

Slide 21 / 138 Slide 21 (Answer) / 138 5 A circular hoop of radius 0.86 m rotates π/3 rad about its center. A 5 A circular hoop of radius 0.86 m rotates π/3 rad about its center. A small bug is on the hoop - what distance does it travel (arc length) small bug is on the hoop - what distance does it travel (arc length) during this rotation? during this rotation? A 0.30 m A 0.30 m B 0.90 m B 0.90 m Answer C 1.4 m C 1.4 m B D 2.7 m D 2.7 m [This object is a pull tab] Slide 22 / 138 Slide 23 / 138 Angular Displacement Angular Displacement Something interesting - look B 3 B 3 at the three concentric B 2 B 2 circles drawn on the rigid Since θ is constant for every disc the radii, arc lengths s r B 1 B 1 point and describes and the points A r and B r . something very fundamental about the rotating disc, it will As the disk rotates, each A 1 A 2 A 3 A 1 A 2 A 3 be given a special name - point A moves to point B, angular displacement. covering the SAME angle θ, but covering a different arc length s r . Thus, all points on a rotating rigid disc move through the same angle, θ, but different arc lengths s, and different radii, r. Slide 24 / 138 Slide 25 / 138 Angular Displacement Angular Displacement We're now in a position to relate angular motion to linear motion. Using the definition of the radian for these three points we show how Linear motion has been covered in the Kinematics section of this the linear displacement any point on the disc increases directly as course. the distance of the point from the axis of rotation (its radius). The radian is now defined. Angular Displacement is defined. But, the angular displacement, θ, remains the same for a constant rotation for all points on the disc. An object at point A 3 rotates to point B 3 , covering a linear displacement of s 3 and an angular displacement of θ, and similarly B 3 for points A 2 and A 3 . B 2 B 3 B 2 B 1 B 1 A 1 A 2 A 3 A 1 A 2 A 3