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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