Physics 1A: Classical Mechanics Fall 2015
Concept Lecture 1-3: Circular Motion
The molecular motor that drives the paramecium flagellum Credit: Protonic NanoMachine Project, ERATO:
http://www.fbs.osaka-u.ac.jp/labs/namba/npn/movies/MotorReversal.mpeg
Ballet Piroutte Credit: Jim Lamberson CC BY-SA 4.0
https://en.wikipedia.org/wiki/Turn_(dance_and_gymnastics)#/media/File:Pirouette.gif
Simulation of interacting galaxies Credit: NASA https://www.youtube.com/watch?v=BwhUl1qvG4k
Hurricane Gonzalo imaged by the NASA GOES satellite Credit: NASA http://www.nasa.gov/downloadable/videos/satellites_see_powerful_hurricane_gonzalo_hit_bermuda.mp4
= arclength
Angular displacement is a dimensionless quantity with units of radians, where 2π radians = 360º
= angular displacement
A
B
units = = 1 meters meters
angular rate
angular speed
angular velocity
A
B
polar coordinate unit vector for angular motion
A → B in time Δt
radial velocity
angular acceleration
A
B
angular acceleration
Constant Circular Motion
x
y
Constant Circular Motion
x
y
Constant angular speed but NOT constant angular velocity
Constant Circular Motion
x
y
Circular motion requires a centripetal acceleration for the change in velocity direction
change in orientation
Children on a roundabout Credit Jayhawksean CC BY-SA 3.0
https://en.wikipedia.org/wiki/File:Merry-go-round.jpg
Acceleration parallel to velocity speeds up or slows down objects
Acceleration perpendicular to velocity turns an object toward a new direct of motion
Helical motion Uploaded by Pieter Kuiper, CC AY-SA 2.0 Generic
https://commons.wikimedia.org/wiki/File:Rising_circular.gif
Deltoid motion Created by Sam Derbyshire, CC AY-SA 3.0
https://commons.wikimedia.org/wiki/File:Deltoid2.gif
Epicycloid motion Created by Sam Derbyshire, CC AY-SA 3.0
https://commons.wikimedia.org/wiki/File:EpitrochoidOn3-generation.gif
Summary
Kinematic quantities of circular motion: angular displacement: Δφ in radians arclength: L = RΔφ angular rate: ω = Δφ/Δt angular speed & velocity: vφ = RΔφ/Δt = Rω
tangent to circle
angular acceleration: α = Δω/Δt, aφ = Rα centripetal acceleration: ac = Rω2 = vφ2/R
Summary
Circular motions are often easier to write in polar coordinates r(t), φ(t) Convert to rectangular using Complex motions seen in nature arise when circular and linear motions are combined