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…
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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
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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
r x y
2
2
  tan ( y x)
1
x  r cos( )
y  r sin(  )
http://en.citizendium.org/images/thumb/1/18/Polar_coordinates_.png/250px-Polar_coordinates_.png
Angular displacement
• Angular displacement – the angular difference
between final and initial angle
  2  1
• Arc length – angular equivalent of distance,
how long the arc is that the angle sweeps out
s  r
Scalar or vector?
Angular velocity
• Angular velocity or angular frequency (ω) –
how much an object’s angular coordinate
changes with time
d

dt
• Related to frequency (f) and period (T)
  2f  2 T
http://en.citizendium.org/images/thumb/1/18/Polar_coordinates_.png/250px-Polar_coordinates_.png
Discussion Questions
• Which way does ball on string go?
Tangentially
• 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?
Ang. Velocity is the same for any point on wheel
Lin. Velocity is greatest at edge
Angular vs. linear velocity

v  rtˆ
v  r
http://edubuzz.org/blogs/advhigherthings/files/2008/09/circle-diagram.jpg
Angular acceleration
• Angular acceleration (α) – how the angular
velocity changes in time
d d 2

 2
dt
dt
• Related to the tangential acceleration (aT)
aT  r
Angular vs. linear acceleration
• Uniform circular motion – α = 0, only
centripetal accel. (aC)
• Non-uniform circular motion – both aT and aC

a  aT tˆ  aC rˆ
aT  r
aT
aC  v   2 r
aC
a  r  2 4
http://upload.wikimedia.org/wikipedia/commons/thumb/2/22/Nonuniform_circular_motion.svg/293px-Nonuniform_circular_motion.svg.png
Angular Kinematics
Angular kinematics – same as linear
• Assume α=constant
1 2
   0   0 t  t
2
   0  t
 2  02  2    0 
Centripetal Force
Centripetal Force
• Force keeping an object moving in a circle

F   maC rˆ
F  maC  m  r
2
http://astronomy.swin.edu.au/cms/imagedb/albums/scaled_cache/centripedal-316x300.png