STARTER
Consider two points, A and B, on a spinning disc.
1. Which point goes through the greatest
distance in 1 revolution?
2. Which point goes through the most
degrees in 1 revolution?
3. Which point has the greatest velocity?
4. Which point has the greatest
angular speed? ( r.p.m.’s)
Practice: Circular Motion
RADIAN MEASURE
q = 1 radian when s = r
s = rq
This is true
when q is
measured in
radians.
180 degrees = p radians
1 radian = 57.3 degrees
1 revolution = 2p radians
Example:
Convert the following:
1. 45 degrees = ________radians
2. 1.5 radians = _________ degrees
3. .8 revolutions = ________ radians
1. 45 degrees (p rad / 180 degrees) = .785 radians
2. 1.5 radians ( 180 degrees / p radians ) = 85.9 degrees
3.
.8rev ( 2 p rad / rev) = 5.03 rads
Angular Velocity
w
Angular velocity is how many radians per second an
object moves through.
w = Dq / Dt
Example:
(rad/s)
A disc spins through 2 revolutions in 3 seconds. What is
the angular velocity of the disc in radians/second?
Solution : Dq = 2 rev ( 2p rads / 1 rev) = 4p rads = 12.6 rads
so w = Dq / Dt = 12.6rads/3 sec = 4.12 rads/sec
Tangential Velocity
v
A point on a disc rotating with an angular
velocity w, has a tangential velocity in m/s.
The velocity of the point depends on how
far it is from the center, in fact:
v = wr
Example:
A disc spins through 2 revolutions in 3
seconds. What is the velocity of a
point 10cm from the disc’s center?
Solution :
v = wr = (4.12 rads/sec )(.10m) = .412 m/s
Angular (w) and Tangential Velocity
(v)
v = wr
For a rotating object, all points have
the same w, but different tangential
velocities.
Angular Acceleration
a
If the angular speed changes with time,
there will be an angular acceleration, a.
a = Dw /Dt
(rad/s2 )
Example: A disc spinning at 10rad/s, slows to 5
rad/s in 2 seconds. What is the angular acceleration?
Solution : a = Dw/Dt = ( 5 - 10) / 2 = -2.50 rad/s2
EXIT
Two children are on a rotating carnival ride. Write a short paragraph
comparing the angular velocity and the tangential velocities of each child.
STARTER
If the chain moves at 1 m/s and the radius of the rear gear
is 8cm, what is the angular speed of the rear gear in rad/s ?
w = v/r = 1/.08 = 12.5 rad/s
Centripetal Acceleration ac
If an object is moving in a circle, it has an
acceleration that points to the center of the
circle, called the centripetal acceleration, ac.
ac =
2
v /r=
2
wr
Example:
A disc spins at 12 rad/s. What is the
centripetal acceleration of a point
10cm from the disc’s center?
Solution :
a = w2r = (12)2(.10) = 14.4 m/s2
Summary
w = Dq / Dt
Angular Velocity
v = wr
Tangential Velocity
a = Dw /Dt
Angular acceleration
ac =
2
v /r=
2
wr
Centripetal
Acceleration
All The Vectors for Rotation
Tangential Velocity ( always there )
Centripetal Acceleration ( always there)
Tangential Acceleration ( only there if its speeding up or slowing down)
Total Acceleration ( the vector sum of at and ac )
Kinematic Equations for Constant
Angular Acceleration
1.
wf = wi + at
2.
Dq = wit + (1/2) at2
Example
A motor starts from rest and
accelerates to 40 rad/s in 10 seconds.
1. What is the angular acceleration?
2. How many radians does the motor turn
through?
To get a, you need an equation with a
in it, but without qf. Which one is it?
1.
wf = wi + at
40 = 0 + 10a or
a = 4.00 rad/s2
To get Dq use 2.
2.
Dq = wit + (1/2) at2
Dq = 0 + (1/2)(4)(42) = 32 radians
Application: Circular Motion
Problem Set
Connection
An audio CD head reads the information from the
disc at a constant rate. This means that the
tangential velocity of the disc where the read
head is must be constant.
This means that as the read head moves closer
to the center of the disc, v = wr = constant.
So as r gets smaller, what must happen to w?
Explain.
EXIT :4 Minute Writing
Carefully describe the difference between angular
velocity and tangential velocity.
Then consider two different points on a
spinning fan blade ( point A is closer to the
center of the blade and point B is near the outer
edge).
Compare their tangential and angular velocities.