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