Transcript Rotational Motion - Damien AP Physics

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?
Radians
• Angular position of
object in degrees=
• More useful is
radians
• 1 Radian= angle
subtended by arc
whose length =
radius
 =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
QuickTime™ and a
TIFF (Uncomp resse d) de com press or
are nee ded to s ee this picture.
• 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: right hand
rule- turn your fingers in
direction of rotation and
if thumb points up=+
– clockwise is – counterclockwise is +
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
Equations of Angular
Kinematics
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LINEAR
a = (vf - vo)/t
vf = vo + at
s = ½(vf + vo)t
s = vot + ½at2
vf2 = vo2 + 2ax
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ANGULAR
α = (ωf - ωo)/t
ωf = ωo + αt
θ = ½(ωf + ωo)t
θ = ωot + ½αt2
ωf2 = ωo2 + 2αθ
Linear vs Angular
They are related!!!
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
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
• Or: =r2
– Towards axis of rotation
• Centripetal force=macentripetal
• If object is not in uniform circular motion,
need to add the 2 vectors of tangential and
centripetal acceleration (perpendicular to
each other) so: a2=ac2+at2
Rolling
QuickTime™ and a
H.264 decompressor
are needed to see this picture.
Rolling
• Rolling= rotation + translation
• Static friction between rolling object and
ground (point of contact is momentarily
at rest so static)
v=r
a=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