Rotational Motion
Rotation of rigid objects- object
with definite shape
A brief lesson in Greek
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 theta
 tau
 omega
 alpha
Rotational Motion
• All points on object
move in circles
• Center of these circles
is a line=axis of
rotation
• What are some
examples of rotational
motion?
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Radians
• Angular position of
object in degrees=ø
• More useful is
radians
• 1 Radian= angle
subtended by arc
whose length =
radius
Ø=l/r
Converting to Radians
• If l=r then =1rad
• Complete circle = 360º so…in a full
circle 360==l/r=2πr/r=2πrad
So 1 rad=360/2π=57.3
*** CONVERSIONS*** 1rad=57.3
360=2πrad
Example: A ferris wheel
rotates 5.5 revolutions. How
many radians has it rotated?
• 1 rev=360=2πrad=6.28rad
• 5.5rev=(5.5rev)(2πrad/rev)=
• 34.5rad
Example: Earth makes 1
complete revolution (or 2rad)
in a day. Through what angle
does earth rotate in 6hours?
• 6 hours is 1/4 of a day
• =2rad/4=rad/2
Practice
• What is the angular displacement of
each of the following hands of a clock in
1hr?
– Second hand
– Minute hand
– Hour hand
Hands of a Clock
• Second: -377rad
• Minute: -6.28rad
• Hour: -0.524rad
Velocity and Acceleration
• Velocity is tangential
to circle- in direction
of motion
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• Acceleration is
towards center and
axis of rotation
Angular Velocity
• Angular velocity = rate of
change of angular position
• As object rotates its angular
displacement is ∆=2-1
• So angular velocity is
 =∆/ ∆t measured in rad/sec
Angular Velocity
• All points in rigid object
rotate with same
angular velocity (move
through same angle in
same amount of time)
• Direction:
– clockwise is – counterclockwise is +
Velocity:Linear vs Angular
• Each point on
rotating object also
has linear velocity
and acceleration
• Direction of linear
velocity is tangent to
circle at that point
• “the hammer throw”
Velocity:Linear vs Angular
• Even though
angular velocity is
same for any point,
linear velocity
depends on how far
away from axis of
rotation
• Think of a merry-goround
Velocity:Linear vs Angular
• v= l/t=r/t
• v=r
Angular Acceleration
• If angular velocity is changing, object would
undergo angular acceleration
• = angular acceleration
=/t
Rad/s2
• Since  is same for all points on rotating
object, so is  so radius does not matter
Angular and Linear
Acceleration
• Linear acceleration has 2 components:
tangential and centripetal
• Total acceleration is vector sum of 2
components
• a=atangential+acentripetal
Linear and Angular Measures
Quantity
Linear
Displacement d(m)
Velocity
v(m/s)
Acceleration
a(m/s2)
Angular
Relationship
Linear and Angular Measures
Angular
Relationship
Displacement d(m)
(rad)
d=r 
Velocity
v(m/s)
(rad/s)
v=r 
Acceleration
a(m/s2)
(rad/s2)
a=r 
Quantity
Linear
Practice
• If a truck has a linear acceleration of
1.85m/s2 and the wheels have an
angular acceleration of 5.23rad/s2, what
is the diameter of the truck’s wheels?
Truck
• Diameter=0.707m
• Now say the truck is
towing a trailer with
wheels that have a
diameter of 46cm
• How does linear
acceleration of trailer
compare with that of the
truck?
• How does angular
acceleration of trailer
wheels compare with
the truck wheels?
Truck
• Linear acceleration is the same
• Angular acceleration is increased because
the radius of the wheel is smaller
Frequency
• Frequency= f=
revolutions per
second (Hz)
• Period=T=time to
make one complete
revolution
• T= 1/f
Frequency and Period
example
• After closing a deal with a client, Kent
leans back in his swivel chair and spins
around with a frequency of 0.5Hz. What
is Kent’s period of spin?
T=1/f=1/0.5Hz=2s
Period and Frequency relate to
linear and angular acceleration
• Angle of 1 revolution=2rad
• Related to angular velocity:
• =2f
• Since one revolution = 2r and the time
it takes for one revolution = T
• Then v= 2r /T
Try it…
• Joe’s favorite ride at the 50th State Fair
is the “Rotor.” The ride has a radius of
4.0m and takes 2.0s to make one full
revolution. What is Joe’s linear velocity
on the ride?
V= 2r
/T= 2(4.0m)/2.0s=13m/s
Now put it together with centripetal acceleration: what is
Joe’s centripetal acceleration?
And the answer is…
• A=v2/r=(13m/s2)/4.0m=42m/s2
Centripetal Acceleration
• acceleration= change in velocity (speed and
direction) in circular motion you are always
changing direction- acceleration is towards
the axis of rotation
• The farther away you are from the axis of
rotation, the greater the centripetal
acceleration
• Demo- crack the whip
• http://www.glenbrook.k12.il.us/gbssci/phys/m
media/circmot/ucm.gif
Centripetal examples
• Wet towel
• Bucket of water
• Beware….inertia is often misinterpreted
as a force.
The “f” word
• When you turn quickly- say in a car or roller
coaster- you experience that feeling of
leaning outward
• You’ve heard it described before as
centrifugal force
• Arghh……the “f” word
• When you are in circular motion, the force is
inward- towards the axis= centripetal
• So why does it feel like you are pushed
out???
INERTIA
Centripetal acceleration and
force
• Centripetal acceleration=v2/r
– Towards axis of rotation
• Centripetal force=macentripetal
Rolling
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Rolling
• Rolling= rotation + translation
• Static friction between rolling object and
ground (point of contact is momentarily
at rest so static)
v=r
Example p. 202
A bike slows down uniformly from v=8.40m/s
to rest over a distance of 115m. Wheel
diameter = 68.0cm. Determine
(a) angular velocity of wheels at t=0
(b) total revolutions of each wheel before
coming to rest
(c) angular acceleration of wheel
(d) time it took to stop
Torque
How do you make an object
start to rotate?
Pick an object in the room and list
all the ways you can think of to
make it start rotating.
Torque
• Let’s say we want to spin a can on the
table. A force is required.
• One way to start rotation is to wind a
string around outer edge of can and
then pull.
• Where is the force acting?
• In which direction is the force acting?
Torque
Force acting on outside of can. Where string leaves the
can, pulling tangent.
Torque
• Where you apply the force is important.
• Think of trying to open a heavy door- if
you push right next to the hinges (axis
of rotation) it is very hard to move. If
you push far from the hinges it is easier
to move.
• Distance from axis of rotation =
lever arm or moment arm
Torque
• Which string will
open the door the
easiest?
• In which direction do
you need to pull the
string to make the
door open easiest?
Torque
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Torque
•  tau = torque (mN)
• If force is perpendicular,  =rF
• If force is not perpendicular, need to find the
perpendicular component of F
 =rFsin
Where = angle btwn F and r
Torque example
(perpendicular)
• Ned tightens a bolt in his car engine by
exerting 12N of force on his wrench at a
distance of 0.40m from the fulcrum. How
much torque must he produce to turn the
bolt? (force is applied perpendicular to
rotation)
Torque=  =rF=(12N)(0.4m)=4.8mN
Torque- Example glencoe p. 202
• A bolt on a car engine needs to be
tightened with a torque of 35 mN. You
use a 25cm long wrench and pull on the
end of the wrench at an angle of 60.0
from perpendicular. How long is the
lever arm and how much force do you
have to exert?
• Sketch the problem before solving
More than one Torque
• When 1 torque acting, angular acceleration
 is proportional to net torque
• If forces acting to rotate object in same
direction net torque=sum of torques
• If forces acting to rotate object in opposite
directions net torque=difference of torques
• Counterclockwise +
• Clockwise -
Multiple Torque experiment
• Tape a penny to each side of your pencil and
then balance pencil on your finger.
• Each penny exerts a torque that is equal to its
weight (force of gravity) times the distance r
from the balance point on your finger.
• Torques are equal but opposite in direction so
net torque=0
• If you placed 2 pennies on one side, where
could you place the single penny on the other
side to balance the torques?
Torque and center of mass
• Stand with your heels against the wall
and try to touch your toes.
• If there is no base of support under your
center of mass you will topple over
Torque and football
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• If you kick the ball at
the center of mass it
will not spin
• If you kick the ball
above or below the
center of mass it will
spin
Inertia
• Remember our
friend, Newton?
• F=ma
• In circular motion:
– torque takes the
place of force
– Angular acceleration
takes the place of
acceleration
Rotational Inertia=LAZINESS
• Moment of inertia for a point object
I = Resistance to rotation
I=mr2 = I 
• I plays the same role for rotational motion as
mass does for translational motion
• I depends on distribution of mass with respect
to axis of rotation
• When mass is concentrated close to axis
of rotation, I is lower so easier to start and
stop rotation
Rotational Inertia
Unlike translational motion, distribution of mass
is important in rotational motion.
Changing rotational inertia
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• When you change
your rotational
inertia you can
drastically change
your velocity
• So what about
conservation of
momentum?
Angular momentum
• Momentum is conserved when no
outside forces are acting
• In rotation- this means if no outside
torques are acting
• A spinning ice skater pulls in her arms
(decreasing her radius of spin) and
spins faster yet her momentum is
conserved
Angular momentum
• Angular momentum=L=mvr
• Unit is kgm2/s
Examples…
• Hickory Dickory Dock…
• A 20.0g mouse ran up a clock and took
turns riding the second hand (0.20m),
minute hand (0.20m), and the hour
hand (0.10m). What was the angular
momentum of the mouse on each of the
3 hands?
• Try as a group.