Slide 1 / 130 Slide 2 / 130 AP Physics C - Mechanics Rotational Motion 2015-12-03 www.njctl.org Slide 3 / 130 Table of Contents Click on the topic to go to that section Rotational Kinematics Review · · Rotational Dynamics · Rotational Kinetic Energy · Angular Momentum
Slide 4 / 130 Rotational Kinematics Review Return to Table of Contents Slide 5 / 130 What is Rotational Motion? Linear Kinematics and Dynamics dealt with the motion of "point particles" moving in straight lines. Life is more complex than that. 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. The player is rotating about his ankle in the first case, and translating (linear) backwards in the second case. Slide 6 / 130 Axis of Rotation In the first case, the player was rotated about his axis of rotation - a line parallel to the ground and through his ankle, 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? "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 7 / 130 Axis of Rotation No - if you were to spin this donut around its center, the axis of rotation would be in the donut hole, pointing out of the page. "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 8 / 130 Rotational Equivalents to Linear (Translational) Motion Linear motion is described by displacement, velocity and acceleration. Rotational motion is described by angular displacement, angular velocity and angular acceleration. It is also important to define the axis of rotation - objects will have different properties depending on this choice. Once the choice is made, it must remain the same to describe the motion. If the axis is changed, then a different rotational motion occurs. Slide 9 / 130 Angular Displacement There are several theories why a circle has 360 degrees. 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, which is the rotational equivalent of displacement (Δs).
Slide 10 / 130 Angular Displacement As a rigid disc (circle) rotates about its center, the angle of rotation, θ, is defined as the ratio of the subtended arc length, s, to the radius of the circle, and is named angular displacement. B A Rotational angles are now defined in geometric terms, not based on an old calendar or arbitrary numbering system. Slide 11 / 130 Radian Angular displacement is unit less since it is the ratio of two distances. But, we will say that angular displacement is measured in radians (abbreviated "rad") . A point moving around the entire circumference will travel 2πr as an angle of 360 0 is swept through. Using the angular displacement definition: B A Slide 12 / 130 Angular Displacement Something interesting - look B 3 at the three concentric B 2 circles drawn on the rigid disc the radii, arc lengths s r B 1 and the points A r and B r . As the disk rotates, each A 1 A 2 A 3 point A moves to point B, 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 13 / 130 1 What is the angular displacement for an arc length (s) that is equal to the radius of the circular rigid body? A 0.5 rad B 1.0 rad C 0.5π rad D 1.0π rad E 1.5π rad Slide 13 (Answer) / 130 1 What is the angular displacement for an arc length (s) that is equal to the radius of the circular rigid body? A 0.5 rad B 1.0 rad Answer C 0.5π rad B D 1.0π rad E 1.5π rad [This object is a pull tab] Slide 14 / 130 2 A record spins 4 times around its center. It makes 4 revolutions. How many radians did it pass? A π rad B 2π rad C 4π rad D 8π rad
Slide 15 / 130 3 A record spins 4 times around its center. It makes 4 revolutions. How many radians did it pass? A π rad B 2π rad C 4π rad D 6π rad E 8π rad Slide 15 (Answer) / 130 3 A record spins 4 times around its center. It makes 4 revolutions. How many radians did it pass? A π rad B 2π rad Answer E C 4π rad D 6π rad E 8π rad [This object is a pull tab] Slide 16 / 130 4 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) during this rotation? A 0.30 m B 0.90 m C 1.4 m D 2.7 m E 3.2 m
Slide 16 (Answer) / 130 4 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) during this rotation? A 0.30 m Answer B 0.90 m B C 1.4 m D 2.7 m [This object is a pull tab] E 3.2 m Slide 17 / 130 5 There are two people on a merry go round. Person A is 2.3 m from 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 π/4. What linear displacement (arc length) is covered by both people? Compare and contrast these motions. Slide 17 (Answer) / 130 5 There are two people on a merry go round. Person A is 2.3 m from 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 π/4. What linear displacement (arc length) is covered by both people? Compare and contrast these motions. Answer The person further from the axis of rotation moves a greater linear displacement, but the same angular displacement as the person closer to the axis of rotation. [This object is a pull tab]
Slide 18 / 130 Other Angular Quantities Angular displacement has now been be related to Linear displacement. Next, let's relate angular velocity and angular acceleration to their linear equivalents: Slide 19 / 130 Other Angular Quantities Define average Angular Velocity (ω avg ) as the change in angular displacement over time, and take the limit as Δt approaches zero for the instantaneous angular velocity (ω). Similarly for average Angular Acceleration (α) and instantaneous angular acceleration (α). Slide 20 / 130 Other Angular Quantities Since ω and α are both related to θ, and θ is the same for all 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. Let's relate the linear velocity and acceleration to their angular equivalents.
Slide 21 / 130 Angular and Linear Velocity Start with angular velocity, and substitute in the linear displacement for the angular displacement. r is constant, so move it outside the derivative 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 22 / 130 Angular and Linear Acceleration Start with angular acceleration, and substitute in the linear velocity for the angular velocity. r is constant, so move it outside the derivative Slide 23 / 130 Angular and Linear Quantities Summary Angular Linear Relationship Displacement Velocity Acceleration
Slide 24 / 130 Angular Velocity sign We're familiar with how to assign positive and negative values to 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 standard convention). B How do we assign a "sign" for angular displacement, velocity or acceleration? A Does "right" or "up" work with rotational motion? Does it work with a circle? Slide 25 / 130 Angular Velocity sign Not really. Not at all. From your math classes, you know that the horizontal axis through a circle is labeled as 0 0 , and angles are measured in a counter clockwise direction. B Once we agree on that, let's look at the definition of angular velocity: A Slide 26 / 130 Angular Velocity sign 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
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