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Centripetal Acceleration
and
Centripetal Force
Acceleration
Speeding up
 Slowing down
 Changing direction
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Think of spinning an object on the end of
a string. How would you describe it to
someone?
 The easiest thing to say is that it moves in
the shape of a circle.
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◦ This means you should be able to measure
the radius of the circle, and from that
calculate things like circumference.
C =2πr
 A = πr2
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You might also notice that since it is going in
a circle, the motion of the bob as it moves
around repeats itself over and over again.
◦ We refer to this as periodic motion,
because periodically ("every so often") it gets
back to where it started and keeps on going.
◦ Other objects, like pendulums, can also be
described as periodic, as long as they repeat the
same motion over and over again.
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The bob is also an example
of two dimensional
motion, just like a projectile.
◦ It is moving in the x and y
directions at the same time,
not in a perfect straight line.
You might remember that when we
originally talked about acceleration we
defined it as a change in the speed and/or
direction of the movement of an object.
 While the bob is going around in a circle,
its speed might be constant
(and usually will be in the
problems you do),
but what is happening
to the direction it is
traveling?
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At any particular instant in time, the velocity
of the bob spinning in a circle is pointing in the
same direction as the tangent line to the circle.
In Figure 1 you can see that the velocity
vectors are pointing in different directions as
the object moves clockwise.
That means that the direction the bob is
traveling is constantly changing!
By definition, this means that the bob
is accelerating.
If it wasn’t accelerating, it would be traveling in
a straight line.
So how do we calculate the acceleration of
the bob?
Centripetal Acceleration
What is the centripetal acceleration of a
person in a car driving at 60km/h in a
traffic circle that is 120m across?
 Change the velocity into metres per
second, and since the measurement of the
circle is a diameter, divide it by two.
 v = 17 m/s
r = 60.0 m
ac= ?
 ac= v 2 / r
= 172 / 60.0
ac= 4.6 m/s2
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Centripetal comes from Greek meaning
“centre seeking”
The bob is basically being pulled in towards
the centre of the circle, which is why it
follows a curved path around the centre. It’s
as though it were seeking the centre of the
circle, even though it never gets there.
Do not confuse this with the word
“centrifugal”, which doesn’t even exist!
◦ A person in a car going around in a circle might say
that a “centrifugal force” was pushing them across
the seat out of the circle.
◦ In fact, it is just your body obeying Newton’s first
law… it is trying to move in a straight line.
◦ The car is trying to keep you going in a circle,
which is centripetal.
Centripetal Force
By now you should be saying to yourself that
if an object going in a circle is accelerating,
there must be a force acting on it.
 This agrees with Newton’s Laws that a force
results in acceleration.
 If there were no forces involved, the object
should be moving at a constant speed in a
straight line, which it isn’t.
 We should be able to come up with some
formulas for forces that relate directly to
circular motion.
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If you have already calculated the centripetal
acceleration of an object moving in a circle
from one of the formulas we have studied so
far, then just go ahead and use F = ma to
calculate the force.
Because this is a force causing an object to
move in a circle, we refer to the force as
a centripetal force.
This is written in formulas as Fc
A lot of the time you won’t calculate the
acceleration first, so we should have a couple
of formulas to be able to calculate the
centripetal force based on the information
you do have.
Centripetal Force
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You might have seen movies with an
astronaut in training spinning around and
around in this big machine to get ready for
their flight. The device is called
a centrifuge, like the one pictured below
(click on it to go to the Nasa website for the Center for Gravitational Biology Research).
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https://www.youtube.com/watch?v=EXQKl-28zn0
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http://videos.howstuffworks.com/discovery/29332-extreme-engineering-acceleration-and-g-force-video.htm
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http://videos.howstuffworks.com/discovery/44485-bad-universe-jet-jockey-video.htm
Let’s look at the forces on a 100 kg
person’s body, and try to relate it back to
regular Earth gravity (one “gee” =
9.81m/s2). Keep in mind that just standing
on the ground the person in this example
would normally weigh (Fg = mg) 9.81e2N.
 What is the centripetal force acting on this
100kg person if he is spun around in a
8.80m circle at…
 10.0m/s?
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Fc = mv2 / r
= (100)(10.0)2 / 8.80
Fc = 1.14e3 N
15.0m/s?
Fc = mv2 / r
= (100)(15.0)2 / 8.80
Fc = 2.56e3 N