Circular Motion
Because, even though we had a lecture on it, it’s still weird.
So last week, we talked about changing from linear motion to circular motion, and
in doing so, we came up with a ton of new ideas and variables. Now, we explained the
origin of all of these in one way, but perhaps that way of explaining wasn’t, in Pokémon
terms, “super-effective.” Now, if it was effective for you, awesome – feel free to skip to
the exercises. If not, then maybe this approach will help. This is what I call the Units
Approach to radians and radial motion.
The Units Approach
Displacement
What is a Radian? A Radian is a ratio of the arc length inside an angle to the
circle’s radius. It looks like this:
Here, the angle measure, in radians, is
  (arc length)/(radius)

s
r
And this measure  is called the angular displacement. From here, we can see
that the radian has no units. This is important! When we derive linear values, we must
add a unit to them, and this means we must multiply by some length unit, as shown
below
 r  s
  r  xtan
where s  xtan is the tangential displacement.
Here we see that in order to get from angular displacement to linear displacement,
we must multiply by the radius r so that we have the proper units for displacement, and
this fits with our definition of angular displacement.
Velocity
It makes sense that, if the angular displacement  is analogous to linear
displacement x, and linear velocity v is defined by
v
x
,
t
then angular velocity (called  ) should be defined by


,
t
and the units for angular velocity are radians per second.
Hang on a minute! You said radians had no units. You’re right – they don’t. So
how in the world could we get from angular velocity to linear velocity? The answer’s
pretty simple here: we have a measure in radians per second (essentially in the units 1 s )
and we need to get to m s . We must, again, multiply by a unit of length. Like we did
before, let’s multiply by r. This will give us meters per second, the unit we need to have
linear velocity.
v  r
Acceleration
In a similar manner, since we know that linear acceleration a is defined by
a
v
,
t
and so it should make sense, at this point, that the angular acceleration  would be
defined by


t
and is measured in radians per second per second, or rad s 2 .
But radians have no units, so in order to get from angular acceleration to linear
acceleration, we have to multiply by a unit of length. We use r again, and this gives
atan 
v r
  

 r 
 r( ) ,
 t 
t
t
and this makes sense, because acceleration a is in m s 2 .
We know of another kind of acceleration, though, and that’s centripetal
acceleration, that keeps objects moving in circles. Now let’s use a definition we already
have to figure out a good transformation for ac in angular terms. We know that
centripetal acceleration ac is defined by
ac 
v2
,
r
and the units here are meters per second per second, or m s 2 , and these units make sense
from the calculation, as shown below
ac 
.
v 2 (m s)2 m 2 s 2 m


 2.
r
m
m
s
Now, we will use a transformation above to find ac in terms of angular velocity
v  r
ac
ac
ac
ac
v2

r
(r )2

r
2 2
r

r
 r 2
and now let us check the units here, using what we know from above.
ac  r 2  m  (1 s)2 
m
s2
And here we can see that the units of acceleration are still m s 2 . Now, it is important to
note that ac is not the same thing as atan , because centripetal acceleration is always
toward the center of the circular path, whereas angular acceleration is around the circular
path.
Exercises
Let’s look at some examples to better acquaint us to these ideas.
1. Calculate the angular velocities of the second, minute, and hour hands of a clock in
radians per second.
2. Calculate the tangential velocity of the arrows on each hand, given that the length of
the second hand is 12cm, the minute hand is 15cm, and the hour hand is 10cm.
3. A unicyclist is traveling at a speed of 6 m s . What is the angular velocity of his feet,
given that his wheel has a radius of 0.5m and his petals have a radius of 0.3m?
4. Ricky swings a yoyo in a loop of radius 0.42m with an angular velocity of 12.56 rad/s.
What is the tangential velocity of the yoyo? What is its centripetal acceleration?
5. If Ricky changes the radius of his loop to 0.84m and its angular velocity to 6.28 rad/s
over the course of 0.5 seconds, what is the new tangential velocity of the yoyo? What is
the new centripetal acceleration?
6. What is the angular acceleration of the yoyo between 4. and 5. What is the tangential
acceleration?
You’ve reached the end! Woo Hoo!
Now for some quick assessments for both you and me.
PIECE OF CAKE.
What parts of this worksheet were the easiest to understand for you? Why do you
think that is?
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STUDY UP
Which parts of this worksheet were the most challenging to you? Why do you
think that is?
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HOW AM I DOING?
This is the part of the assessment where you guys let me know what things I’m
doing well, what things I could do better, and what kinds of things you would like to see
more of.
KEEP
What am I doing well as your SLA Facilitator?
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CHANGE
What things could I do better?
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HELP ME OUT
What are some ideas for practice that would help you learn the material better, or
make you feel more comfortable with it?
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