Circular Motion
Circular Motion
• Can an object accelerate if its speed remains
constant?
• Yes, b/c velocity is a vector quantity; just as
any change in speed means that there is a
change in velocity, so too does a change in
direction indicate a change in velocity.
Rotation & Revolution
• Axis – straight line around which
rotation takes place
• Rotation – the spinning motion
that takes place when an object
rotates about its own axis
(ballerina, ice dancer, etc…)
• Revolution – Motion of an object
turning around an axis outside the
object. (marker on the record, you
on the earth, etc…)
The Earth
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What does the earth do, revolve or rotate?
BOTH!!
It revolves around the sun
It rotates about its axis (north-south pole axis)
Radians, AAAahhh!!
• What is a radian?
• It is when the arc on the
outside of a circle is equal
in length to the circle’s
radius
• A Useful Conversion
1 rev. = 2π(rad) = 360 ˚
From degrees to radians
Convert the following into radians
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35 ˚
175 ˚
235 ˚
360 ˚
465 ˚
520 ˚
720 ˚
Take 5 & Solve!!
From degrees to radians
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Convert the following into radians
35 ˚
= 0.61 rad
175 ˚
= 3.05 rad
235 ˚
= 4.10 rad
360 ˚
= 6.28 rad
465 ˚
= 8.12 rad
520 ˚
= 9.08 rad
720 ˚
= 12.57 rad
‘s
Radians and Arc Length Lab
• Radians and Arc Length
Lab
• Students will be able to
develop the relationship
between radian and arc
length
• Students will be able to
demonstrate and explain
why there are 2π radians
in one full revolution.
PROVE IT!! - LAB TIME!!!!
Lab Time!
The Speeds?
• Linear Speed – is simply distance covered divided
by time. It is what we have called “speed”
v = distance / time
• Angular Speed – is simply the angle covered
divided by time. It is the number of rotations per
unit of time.
ω = Number of Radians / time
(Record Demo)
Angular Speed
Question:
• On a merry-go-round, where would you be
moving with the greatest linear speed?
Rotational Speed?
Answer:
• Outside
• Anywhere
Angular Speed (ω )
How to Measure Angular Speed:
To calculate the angular speed angle (in radians)
and the time for one revolution
ω = 2π(rad) / T
The time for an object to make one revolution is
called the period (T)
From Angular speed to Linear
• To calculate the linear speed of an object in
circular motion we need to know the angular
speed and the radius.
• Together:
• V = ωr
• Angular velocity x radial distance
• The further you are from the center of a rotating
platform, the greater your linear velocity.
Angular Velocity Sample Problems
*solve using radians / second
Solutions
• 1 revolution: (2π) / 4.56 s = 1.38 rad/s
• 1 revolution: (2π)/ 24 hr (86,400 s) = 7.3 x 10-5 rad/s
• 3.25 revolutions: 3.25(2π) / 1 s = 20.4 rad/s
• 45 revolutions = 282.7 rad
282.7 rad / 60 s = 4.7 rad/s
So, What is the Linear speed
V=ωr
• Merry Go Round:
1.38 rad/s x 8.5 m = 11.73 m/s
• Earth:
7.3 x 10-5 rad/s x (6.38 x 106m) = 463.7 m/s!!!
That means 1038 mph!!
*Earth radius is 6.38 x 106m
*This means you are going 463.7 m/s… right now!!!