PHYS16 – Lecture 21

Circular Motion October 27, 2010

Administration

• Test– Th/F 1-4 pm – Merrill 200– Same Format

• Remember 4 assignments due Sunday or Monday

Where have we been…

• Position, Velocity, and Acceleration• 1D and 2D Motion – Kinematics• Force – Newton’s laws• Momentum• Energy

http://www.teachengineering.org/collection/cub_/lessons/cub_images/cub_rockets_lesson02_figure3.jpg

Where we are going…

• Applications– Circular Motion and Rotation– Static Equilibrium– Gravitation and Movement of Planets– Fluids

• Mechanical waves

http://www.ux1.eiu.edu/~cfadd/1350/06CirMtn/Images/loop.gif

Circular Motion and Rotation

• Angular disp., velocity, and acceleration• Circular motion kinematics• Centripetal force• Angular momentum• Torque and inertia

• Simple Machines II – gears, belts, and levers

Angular displacement, velocity, and acceleration

Polar Coordinates

http://en.citizendium.org/images/thumb/1/18/Polar_coordinates_.png/250px-Polar_coordinates_.png

)sin(

)cos(

)(tan 1

22

ry

rx

xy

yxr

Angular displacement

• Angular displacement – the angular difference between final and initial angle

• Arc length – angular equivalent of distance, how long the arc is that the angle sweeps out

12

rs

Scalar or vector?

Angular velocity

• Angular velocity or angular frequency (ω) – how much an object’s angular coordinate changes with time

• Related to frequency (f) and period (T)

http://en.citizendium.org/images/thumb/1/18/Polar_coordinates_.png/250px-Polar_coordinates_.png

dt

d

Tf 22

Discussion Questions

• Which way does ball on string go?

• If you have a rotating wheel and I want to velcro a marble to it, where should I velcro to get max ang. velocity? Where should I velcro to get max lin. velocity?

Tangentially

Ang. Velocity is the same for any point on wheelLin. Velocity is greatest at edge

Angular vs. linear velocity

rv

trv

ˆ

http://edubuzz.org/blogs/advhigherthings/files/2008/09/circle-diagram.jpg

Angular acceleration

• Angular acceleration (α) – how the angular velocity changes in time

• Related to the tangential acceleration (aT)

2

2

dt

d

dt

d

raT

Angular vs. linear acceleration

• Uniform circular motion – α = 0, only centripetal accel. (aC)

• Non-uniform circular motion – both aT and aC

http://upload.wikimedia.org/wikipedia/commons/thumb/2/22/Nonuniform_circular_motion.svg/293px-Nonuniform_circular_motion.svg.png

aC

aT

42

2

ˆˆ

ra

rva

ra

rataa

C

T

CT

Angular Kinematics

Angular kinematics – same as linear

• Assume α=constant

020

2

0

200

2

2

1

t

tt

Centripetal Force

Centripetal Force

• Force keeping an object moving in a circle

rmmaF

rmaF

C

C

2

ˆ

http://astronomy.swin.edu.au/cms/imagedb/albums/scaled_cache/centripedal-316x300.png