PHYS16 – Lecture 22
http://xkcd.com
Circular Motion and Rotation
October 29, 2010
Circular Motion and Rotation
• Circular Motion
– Angular disp., velocity, and acceleration
– Centripetal force
– Circular motion kinematics
• Rotation
–
–
–
–
Inertia
Kinetic Energy
Angular momentum
Torque
• Simple Machines II – gears, belts, and levers
Circular Motion and Centripetal Force
last time…
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, velocity, and
acceleration
• Angular displacement –   2  1
• Arc length – s  r
d v
2
  2f 
• Angular velocity (ω) –  
dt
r
T
d d 2 aT
 2 
• Angular acceleration (α) –  
dt
dt
r


ˆ
ˆ
a

a
t

a
r
T
C
• Linear acceleration( a) –
aT  r
aC  v   r
2
Angular displacement, velocity, and
acceleration
• Uniform circular motion – α = 0, only
centripetal accel. (aC)
• Non-uniform circular motion – both aT and aC
aT
aC
http://upload.wikimedia.org/wikipedia/commons/thumb/2/22/Nonuniform_circular_motion.svg/293px-Nonuniform_circular_motion.svg.png
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
Practice Question #1
• What is the anglular displacement of the Earth
during winter? Assume winter is ¼ of the year
and that the Earth’s orbit can be
approximated by a circle.
A)
B)
C)
D)
0.785 rad
1.57 rad
3.14 rad
None of the above
Practice Question #2
• A CD is spinning with a constant angular
velocity. From last time we know that the
linear velocity of a point on the edge of the CD
is greater than a point in the middle. What
about the centripetal acceleration?
A)
B)
C)
D)
Centripetal acc. is greater at the edge
Centripetal acc. is greater in the middle
Centripetal acc. is equal for both points
There is not enough information
Discussion Question
• I have a ball attached to a string that runs
through a tube and is attached to a mass at
one end. I hold the tube and rotate the ball in
uniform circular motion while holding the
mass. Then I let the mass go. What happens to
the angular velocity of the ball?
(Increase, Decrease, Stay the same)
Circular Motion Kinematics
Angular kinematics – same as linear
• Assume α=constant
1 2
   0   0 t  t
2
   0  t
 2  02  2    0 
Example Question: The Centrifuge
• A centrifuge rotates with an angular speed of 3600
rpm. Then it is switched off and it rotates 60 times
before coming to rest. What was the angular
acceleration that made it stop?
 2  02  2
 02
 (3600 * 2 / 60) 2


2 
2 * 60 * 2 
  188 rad/s  200 rad/s
2
http://upload.wikimedia.org/wikipedia/commons/0/0d/Tabletop_centrifuge.jpg
2
Example Question: The Discus
• A discus thrower with arm radius of 1.2 m starts from
rest and then starts to rotate with an angular
acceleration of 2.5 rad/s2. How long does it take for
the throwers hand to reach 4.7 rad/s?
   0  t
t  (  0 ) /   4.7 / 2.5
t  1.9 s
Rotation
Rotational Inertia
• In linear motion we just care about mass
• In rotational motion we care about how mass
is distributed so we need rotational inertia (I)
n
I   mi ri 2
i 1
formulas pg. 320
• Which has more rotational inertia?
A)
B)
A does!
Rotational Kinetic Energy
• For rotational kinetic energy we use rotational
inertia instead of mass and angular velocity
instead of linear velocity
1 2
K  I
2
• What is kinetic energy of the earth? Mass =
5.98E24 kg and radius=6.37E6 m.
1 2 1 2
2
2
K  I  ( M earthRearth )(
)  2.6E29 J
2
2 5
1 day
Example Question: Rolling vs. Sliding
• Which has more energy: a cylinder that slides
down a ramp with a speed of v0 or a cylinder
that rolls down a ramp with the same speed?
A)
B)
C)
D)
Cylinder that slides
Cylinder that rolls
Both are equal
There is not enough information
1 2 1 2
1 2
K rolling  mv  I  (1  c) mv
2
2
2
Angular Momentum
• Angular momentum (L) – momentum of a
rotating object
  
Lrp
L  rp sin(  )  I
• Cross product like the dot product is a way to
multiply vectors, except cross product gives
vector not scalar
• Direction of cross product is given by right hand
rule
Example Question: Ice Skating
• In a spin, why do ice skaters decrease their
angular velocity when they hold their arms
out?
Kristi Yamaguchi
L=Iω
Holding arms out increases I.
If L stays the same,
and I increases then
ω decreases.
http://www.corbisimages.com/images/67/7760610C-6DF3-4A39-ACD6-C3CDEFF73296/PN015983.jpg
Torque
• Torque (τ) – a force that acts at a distance,
usually causing rotation
  
  r F
  rF sin(  )  I
• Units = Joules
• Vector quantity, direction given by right hand
rule
Example: Jet turbine
• The turbine of a jet engine has a moment of
inertia of 25 kg∙m2. If the turbine is
accelerated uniformly from rest to an angular
speed of 150 rad/s in a time of 25 s, what is
the torque?
   0  t
  (  0 ) / t
  I  I (  0 ) / t  (25)(150) / 25  150 J
Main Points
• Parameters for circular motion/ rotation
basically have linear equivalents
– θ is related to x, ω is related to v, α is related to a
– I is related to m
– L is related to p, L=Iω=rpsin(θ)
– τ is related to F, τ=Iα =rFsin(θ)