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Circular Motion
What is circular motion?
• Objects that move in a circle
experience circular motion.
• I know that’s tough.
• Let’s take a moment and let
it sink in…
Now that that is out of
the way…
• There are specific features
of circular motion that make
it different from linear or
projectile motion
Constant speed
• An object with a constant
speed, which experiences
no other forces, will travel in
a straight line at that speed
infinitely
• Newton’s First Law
Constant speed
Constant speed
• Objects can travel in a circle
and maintain a constant
speed
Is velocity constant?
• No
• Velocity is speed in a given
direction
• Those directions cannot be
circular
What is acceleration?
• Acceleration is a change in
velocity
• We have so far defined
acceleration as a change in
speed but it can be a
change in direction, also
Circular Motion
• An object traveling in a
circle travels at a constant
speed but is accelerating
What causes
acceleration?
• All changes in velocity are
caused by a force
• F = ma
• Newton’s Second Law
What is the force?
• The force keeping the object
in its circular path is called a
centripetal force
• Centripetal means “center
seeking”
Centripetal force
• It is a real force
• It is a contact force
What direction does it
point?
• The centripetal force always
points towards the center of
the circle
What applies the force?
• It depends on the situation
• In general, whatever keeps
the item in it’s circular path
applies the centripetal force
Example
Example
Example
Are there other forces?
• When you make a turn in
your car, what makes you
pull to one side?
• When you swing a bucket of
water above your head,
what keeps the water in the
bucket?
What causes that?
• In truth, it is a delicate
interplay between the inertia
of the item and the
acceleration
• It is another force
Centrifugal force
• From the Latin, centrum,
“center,” and fugere,
“fleeing”
• This is the force that pushes
away from the center of the
circle
Centrifugal Force
• It is the reaction force that
compliments the action of
the centripetal force
• Newton’s Third Law
Centrifugal
• The centrifugal force is a
fictitious force
• Is it also a contact force
Example
• You have a bucket of water
and you are swinging it
around above you head.
What forces are acting on it
and what do they act on?
The two forces
• Remember, we have two
forces, the centripetal and
the centrifugal
• The centripetal acts on the
bucket
• The centrifugal acts on the
water
The math
• You knew it was coming
• Math is the language of
physics and you need to
learn to speak that language
Centripetal acceleration
• There are two equations we
can use depending on what
we know
The first (and easiest)
2
v
ac 
r
• v is the velocity of the object
• r is the radius of the circle
The second
4 r
ac  2
T
2
• T is the time it takes for one
full revolution
• r is the radius of the circle
Centrifugal acceleration
• If the centrifugal force arises
from Newton’s Third Law
and is the equal but
opposite reaction to the
centripetal force, what is the
equation going to be?
Centrifugal acceleration
4 r
ac   2
T
Or
2
v
ac  
r
2
Sample problem
• A 1000 kg car enters an 80
meter radius curve at 20
m/s. What centripetal force
must be supplied by friction
so the car does not skid?
What do we know?
• m = 1000 kg
• r = 80 m
• v = 20 m/s
Find the force
• F = ma =
• F = 1000 × (202/80)
• F = 1000 × 400/80
• F = 1000 × 5 = 5000 N
2
mv /r
Sample problem
• The centripetal force on a
0.82 kg object on the end of
a 2.0 m massless string
being swung in a horizontal
circle is 4.0 N. What is the
tangential velocity of the
object?
What do we know?
• m = 0.82 kg
• r = 2.0 m
• Fc = 4.0 N
Find the velocity
• F = ma =
• 4.0 = 0.82 × v2/2.0
• 8.0 = 0.82v2
• v2 = 9.76
• v = 3.12 m/s
2
mv /r
Sample problem
• A dragonfly is sitting on a
merry-go-round 2.8 m from
the center. If the centripetal
acceleration of the dragonfly
is 3.6 m/s2, what is the
period of the merry-goround?
What do we know?
• r = 2.8 m
• a = 3.6 m/s2
Find the period
• ac =
• 3.6 = (4π2 × 2.8)/T2
• 3.6 = 110/T2
• T2 = 31
• T = 5.5 s
2
2
(4π r)/T
Sample problem
• A car moving at a 1.08 ×
m/s (30 km/h) rounds a
bend in the road with a
radius of 21.2 m. What is
the centripetal acceleration
on the car and the
centrifugal acceleration on
the occupants?
8
10
What do we know?
• v = 1.08 ×
• r = 21.2 m
8
10
m/s
Centripetal
•a=
• a = (1.08 × 108)2 / 21.2
• a = 5.50 × 1014 m/s2
2
v /r
Centrifugal
•a=
• a = -(1.08 × 108)2 / 21.2
• a = -5.50 × 1014 m/s2
2
-v /r