Chapter 16. circular motion (part 1)
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limber up
Limber up for the Kentucky Hamster Derby
Universally acclaimed as the most exciting two minutes in a
wheel, the Kentucky Hamster Derby is big business! You’ve
been called upon by one of the biggest owners in the business
to implement an exacting training programme that’s been
tailored for the big race.
At the moment, the hamsters aren’t following their schedule.
Some are slacking off, and others are over-working. It’s up to
you to make sure that the hamsters train exactly as they should.
Hey kiddo, this Kentucky Hamster
Derby is big business, and we gotta get
the training schedule absolutely spot on!
Distance
(km)
Speed
(km/h)
15.0
3.0
10.0
4.0
2.0
5.5
Total number
of revolutions
Motor setting
)
(
Billionaire
hamster owner
Hamsters like to run all night in
their wheels. The training schedule
is a way of honing the kinds of
distances and speeds they typically
run at into training for a race.
Distance
(km)
Speed
(km/h)
15.00
3.00
10.00
4.00
2.00
5.50
Total number
of revolutions
Motor setting
(
)
The hamsters are to do three different types of run, depending
on which day of their schedule they’re on. The hamster owner
knows the distances he’d like the hamsters to cover and the
speeds he’d like them to run at.
But he’s not great at physics - which is where you come in!
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circular motion (part 1)
You can revolutionize the hamsters’ training
It’s time to design the ultimate hamster training tool.
The equipment you have to achieve this is:
A standard hamster racing wheel with a
radius of 10.0 cm. This means that it
is 10 cm from the center of the wheel to
the edge, where the hamster runs.
End of motor can
be attached to
wheel to turn it.
A motor which attaches to
the wheel to make it turn.
It has some numbers on it,
and higher numbers make
the motor turn faster - but
unfortunately the units have
been rubbed off.
10.0 cm
The radius is
the distance
from the center
to the edge.
Motor has numbers on but unfortunately the units
have been rubbed off.
0 5 10 15 20 25
Motor turns faster when
slider moved to the right.
A counter that counts wheel
revolutions. You can use this
to start the motor, then stop
it after a certain number of
wheel revolutions.
Counter can be
programmed to
stop motor after
a certain number
of revolutions.
The schedule involves linear distances and
speeds, which you’d usually measure along a
straight line or by using component vectors.
Counter keeps track
of the number of
wheel revolutions.
But the hamster wheel is circular, and
the counter keeps track of the number of
revolutions. How are we going to go from
the linear schedule to the circular equipment?
Circular motion
is different from
linear motion.
The schedule is linear, but the
equipment is circular. What are
you going to do next?
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circumference and revolution
So, there’s big bucks
in this hamster
racing thing then!
Joe: Yeah - it’s really important to get this training schedule
automated and set up!
Jim: So I guess we use the motor to make the wheel turn at a
certain speed, and have the timer switch it off when it’s covered
the right distance? That sounds pretty straightforward.
Frank: We already know the wheel’s 10.0 cm all the way round ...
Jim: No - the radius of the wheel is 10.0 cm, but that’s the
distance from its center to its edge. We need to know the distance
the outside of the wheel travels in one revolution.
Circumference is a
word that means
the ‘perimeter’ of
a circle.
The circumference
of a circle is
the DISTANCE
something at the
edge travels in one
REVOLUTION.
You can also think of the
circumference as the distance
around the edge of a stationary
circle, but in physics it’s more
useful to think about it in terms
of a rotating circle.
Frank: Yeah, OK, so we need to figure out the distance a hamster
runs for each revolution of the wheel. The counter counts the
number of revolutions the wheel’s made. So we work out the
number of revolutions required to cover each distance, and we’re
fine.
Joe: But if we don’t know the circumference of the wheel, we
can’t do anything with that.
Frank: What’s a circumference?
Joe: The circumference is the special name for the perimeter of a
circle - the distance all the way round the outside.
Jim: OK, so we need some way of working out the circumference.
If we know what distance the hamster covers when the wheel goes
round once, we can use that to work out how many revolutions
the wheel needs to do to cover each distance.
Jim: And if we know that, then we can set the counter to turn the
motor off once the wheel’s done the correct number of revolutions.
Frank: I guess we need to work out what those numbers on the
motor mean as well, so we can get the speed right. It’s a shame the
units got rubbed off.
Joe: Yeah, that’s a good point.
Jim: Well, speed is distance divided by time, isn’t it? So if we get
the distance sorted out first, we can think about setting the right
speed later.
Frank: Cool, let’s go for it!
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circular motion (part 1)
Thinking through different approaches helps
Wheel is a circle.
You need to work out the hamster wheel’s circumference
(the distance all the way round the edge) so that you know
how many revolutions are equal to each distance in the
training schedule.
But all you know at the moment is that the wheel is a circle
with a 10.0 cm radius (the distance from its center to its
edge). How are you going to work out its circumference so
that you can implement the training schedule?
Use the circumference (once
you know it) to work out how
many revolutions the wheel
needs to do for each distance.
circumference = ?
10.0 cm
radius = 10.0 cm
You have a hamster wheel and want to know the distance that the hamster will cover when the wheel
goes round once - i.e., the circumference of the wheel (which has a 10.0 cm radius).
Write down as many methods as you can think of to work this out.
You’re describing practical methods, not giving a numerical answer.
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equations save time
v
You have a hamster wheel and want to know the distance that the hamster will cover when the wheel
goes round once - i.e., the circumference of the wheel.
Write down as many methods as you can think of to work this out.
You’re describing practical methods, not giving a numerical answer.
Get a piece of string, wrap it
once round the wheel.
Mark where it comes to, then
take the string off the wheel and
measure it with a ruler.
Remove the wheel from the stand and make a mark
on it. Line up this mark with a mark on the ground.
Then roll the wheel along until the mark is touching
the ground again.
Measure the distance between the first mark on
the ground and the second mark on the ground.
It’d be a shame to have to do all that again
if we decide to use a different wheel, or if we
have to deal with other circles in the future. Is
there an equation we could use instead?
Equations save you time.
You can work out the distance around the
hamster wheel by wrapping string around
it or rolling it. But what if you need to deal
with other circles in the future?
It’s definitely easier to use a ruler to
measure a linear distance (like the radius)
than it is to measure a curved distance (like
the circumference), so an equation linking
the radius and circumference would be
useful!
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If you have to
do a task more
than once, try
to work out an
equation that’ll
save you time.
circular motion (part 1)
A circle’s radius and circumference are linked by π
All circles are similar. They’re exactly the same shape zoomed in or out.
Although they don’t have ‘sides’, you can be sure that the ratio of a
circle’s circumference to its radius will always be the same. (Just
like the sides of similar triangles always have the same ratios.)
A circle’s radius fits around
its circumference:
...twice...
r
...three times...
A circle’s radius fits around its circumference
approximately 6.28 times (3 sd).
r
r
Once...
Radius = r
...four times...
0.28r
r
r
...6.28 (3 sd)
times in total.
...six times...
r
...five times...
Circumference
This is another way of saying that
whatever size the circle is, its radius
will always fit around its circumference
the same number of times.
Radius
C = 2πr
π = 3.14 (3 sd)
The actual ratio of the lengths is a number
with an infinite number of significant digits!
So rather than writing “6.28 (3 sd)” as the
ratio, there’s a special abbreviation for it - 2π,
where π = 3.14 (3 sd).
So you can write down the equation C = 2πr,
where C is the circumference and r is the
radius. In other words, if a circle’s radius is
1.00 m, its circumference is 6.28 m, and so on.
But that’s just dumb! Why not make π
twice as big, i.e. π = 6.28... , so that the ratio is
π instead of 2π? If I get to make up a new number,
surely it’s easier not to have that 2 in there!
In physics, the
RADIUS is more
interesting than
the diameter.
π was invented when the diameter
was more interesting than the radius.
Mathematicians originally defined π as the ratio
between a circle’s diameter (the distance across the
circle) and its circumference. The diameter is twice
the length of the radius, so π = 3.14 (3 sd) as the
diameter fits round the circumference 3.14 (3 sd)
times. Once π has been defined, you can’t really go
changing its value.
For example,
torque = radius x force
In physics, you’re often a lot more interested in the
radius of a circle than you are in its diameter (for
example, torque = radius × force), so the value 2π
will often turn up in your equations.
π = 3.14 (3 sd)
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� is a ratio
Q:
A:
Q:
What are the units of π ?
You said that π has an infinite number of digits?
Surely it must end somewhere!
π is a ratio of two lengths, the circumference and the
diameter. It tells you the NUMBER of times the diameter fits
into the circumference - 3.14 (3 sd) times. Length divided by
length is dimensionless, so π is a NUMBER and doesn’t have
any units.
Q:
A:
Do I need to remember the value of π for my exam?
Knowing that π = 3.14 (3 sd) is useful. If you’re doing
AP Physics, you won’t have a calculator in the multiple choice
section, so the approximation π = 3 (1 sd) will help you to
choose the option that’s in the right ballpark.
Q:
Can I use the [π] button on my calculator in the
section where I’m allowed a calculator?
A:
Q:
Yes, that’s be fine - though you need to round your
answer to an appropriate number of significant digits at the end.
But why is π 3.14 (etc) in the first place, when that
makes the ratio of a circle’s radius to its circumference 2π?
A:
π was originally coined to describe the ratio between a
circle’s circumference and its diameter: C = πd. Nowadays, the
ratio of a circle’s circumference to its radius is more often used
in physics - but the value of π had already been decided.
The ratio of a circle’s circumference to its diameter is π. As
the radius is half the length of the diameter, the ratio of the
circumference to the radius is 2π. So C = 2πr
Radius = r
C = 2πr
Diameter, d= 2r
C = πd
A:
Q:
Nope! π is an irrational number, which means you can’t
write it down exactly.
Is that why there’s a symbol for it then? To avoid
having to write more of it out than you really need to?
A:
That’s right. You can write the equation:
circumference = 2πr and the ‘=’ sign is true because when you
use the symbol π it implies the full irrational number!
Otherwise you’d have to write circumference = 6.28r (3 sd)
as you’d never be able to write down the exact value of the
circumference as an equation.
Q:
That all sounds a bit philosophical to me. I guess
that in practice, I can write down circumference = 2πr
when I’m showing my work, then at the very end I can use
the value 3.14 (3 sd) for π when I’m actually putting the
numbers in to work out an answer?
A:
Q:
That’s a very good way of thinking about it.
now?
A:
So do I get to use π to design the hamster trainer
On you go then ...
π is the ratio of two lengths
- it tells you the NUMBER of
times the diameter fits into
the circumference. Because π
is a NUMBER, it has no units.
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circular motion (part 1)
Convert from linear distance to revolutions
The hamster training schedule contains
distances in km. Your job is to implement the
schedule using the wheel and a counter that
keeps track of the number of revolutions. So
you need to work out how many circumferences
- and therefore how many revolutions - each
distance is equivalent to.
After 1 revolution,
a hamster has run
a distance of 1
circumference.
Don’t worry about
this column yet,
you’ll fill it in later.
a. Assume you have a hamster wheel, radius r.
Write down an equation in terms of r that gives
you the distance that something on the edge of
the circle will cover if the circle rotates once.
c. Your hamster wheel has a radius of 10.0 cm. Fill in the
‘total number of revolutions’ column for the distances
shown in the hamster training schedule. There’s space
under the table for you to show your work.
Distance
(km)
Speed
(km/h)
15.00
3.00
10.00
4.00
2.00
5.50
Total number
of revolutions
Motor setting
(
)
b. You would like the hamster to cover distance x.
Write down an equation that gives you x in terms
of r. and the total number of revolutions, N.
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check your units
Don’t worry about
this column yet,
you’ll fill it in later.
a. Assume you have a hamster wheel, radius r.
Write down an equation in terms of r that gives
you the distance that something on the edge of
the circle will cover if the circle rotates once.
r
Circumference C = 2πr
b. You would like the hamster to cover distance x.
Write down an equation that gives you x in terms
of r. and the total number of revolutions, N.
c. Your hamster wheel has a radius of 10.0 cm. Fill in the
‘total number of revolutions’ column for the distances
shown in the hamster training schedule. There’s space
under the table for you to show your work.
Distance
(km)
Speed
(km/h)
Total number
of revolutions
15.00
3.00
23900 (3 sd)
10.00
4.00
15900 (3 sd)
2.00
5.50
3180 (2 sd)
Motor setting
(
)
Number of revolutions is the same as the number
x
x = 2πrN
N = 2πr
of circumferences in distance x:
r = 0.100 m
3
x
15.00 km: N = 15.00 × 10
N = C
and
C = 2πr
2 × π × 0.100
If you have a mixture of distance
x
units, it’s usually safest to convert N = 23900 revolutions (3 sd)
N = 2πr
everything to meters.
10.00 × 103
10.00
km:
N
=
x = 2πrN
2 × π × 0.100
How do you get units
N = 15900 revolutions (3 sd)
of revolutions from a distance
divided by a distance? Doesn’t that
fly in the face of everything we’ve said
about units up until now?
× 10
2.00 km: N = 2 ×2 .00
π × 0.100
N = 3180 revolutions (3 sd)
Your answer is a NUMBER of revolutions.
To work out the number of wheel revolutions in 15
km, you divide the total distance (15000 m) by the
circumference of the wheel (2πr = 0.628 m). This gives
you an answer of 23900 - and there are no units, as a
length divided by a length is dimensionless.
So be careful if you’re checking the units of an
equation or answer. A quantity that represents
a number (of things, for example number of
revolutions) doesn’t have units!
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3
The units divide
out and cancel.
15000 m
= 23900 (3 sd)
0.628 m
The final answer
is a NUMBER
with no units.
circular motion (part 1)
The distances are cool - but
the speeds are really important
too. I think the motor might be
in Hertz ... can you help?
Convert the linear speeds into Hertz
The distances slot into the training schedule brilliantly - but
the speed the hamsters train at for each session is also vital.
You have a motor you can use to turn the wheel ... but the
units have been rubbed off. However, the hamster owner
thinks that the motor might be marked in Hertz.
Hertz is always
capitalized, and can be
abbreviated to Hz.
Hertz is a unit of frequency that describes the number of times
per second something regular happens. In physics, this is often
referred to as the number of cycles per second.
So if the wheel goes round 5 times per second, you can say that it
has a frequency, f, of 5 Hz.
The period of the wheel is the time the wheel takes
to do one rotation, and is given the symbol T.
You can calculate the period from the frequency. If
something happens 5 times per second (so has a
frequency of 5 Hz), then it happens 5 times in 1 second.
So its period must be T = 1 = 0.2 s.
5
a. What time, in seconds, does it take the
hamsters to cover 15.00 km at 3.00 km/h?
b. You already worked out that this training plan
involves the wheel doing a total number of 23900
revolutions. Calculate the period, T, and hence the
frequency, f, of the wheel, and fill in the table.
Frequency
f =
1
T
Period
Motor setting
)
(
Total number of
revolutions
Distance
(km)
Speed
(km/h)
15.0
3.0
23900 (3 sd)
10.0
4.0
15900 (3 sd)
2.0
5.5
3200 (2 sd)
Period
T =
1
f
Frequency
Distance
(km)
Speed
(km/h)
Total number
of revolutions
15.00
3.00
23900 (3 sd)
10.00
4.00
15900 (3 sd)
2.00
5.50
3180 (3 sd)
Motor
frequency (Hz)
c. Do similar calculations to fill in the rest of the table.
The period, T, is the time for
one rotation of the wheel.
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answers and questions
Distance
(km)
Speed
(km/h)
Total number
of revolutions
Motor
frequency (Hz)
15.00
3.00
23900 (3 sd)
1.33 (3 sd)
10.00
distance
speed = distance
time
=
time
speed
2.00
15.00 km = 5 hr
time = 3.00
km/hr
min × 60 s = 18000 s (3 sd)
5 hr = 5 hr × 601 hr
1 min
4.00
15900 (3 sd)
1.77 (3 sd)
5.50
3180 (3 sd)
2.43 (3 sd)
a. What time, in seconds, does it take the
hamsters to cover 15.00 km at 3.00 km/h?
b. You already worked out that this training plan
involves the wheel doing a total number of 23900
revolutions. Calculate the period, T, and hence the
frequency, f, of the wheel, and fill in the table.
Period is time for 1 revolution.
23900 revolutions take 18000 s.
18000 = 0.753 s (3 sd)
T = 23900
1
1
f = T = 0.753 = 1.33 Hz (3 sd)
Q:
What’s the difference between the
frequency and the period?
A:
The frequency is the number of cycles
per second - the number of times a regular
thing happens per second.
The period is the number of seconds per
cycle - the number of seconds that it takes
for a regular thing to happen once.
Q:
Why is frequency measured in
Hz instead of however you’d write “per
second” as a unit?
A:
“Hz” is easier to write than “1/s” (which
is the correct format) and less open to being
misinterpreted.
c. Do similar calculations to fill in the rest of the table.
10.00 km at 4.00 km/h takes 10.00
4.00 × 60 × 60 = 9000 s.
9000
1
15900
T = 15900
f = T = 9000 = 1.77 Hz (3 sd)
2.00 km at 5.50 km/h takes 2.00
5.50 × 60 × 60 = 1310 s.
1310
1
3180
T = 3180
f = T = 1310 = 2.43 Hz (3 sd)
This is a shortcut that stops you
having to calculate a value for T
before you calculate a value for f.
Q:
I’ve seen frequency written in units
of s in other books. Is that OK?
-1
Q:
What if the regular thing happens
less than once per second?
A:
A:
Q:
Q:
A:
Yes, that’s fine, and is another way of
representing units using scientific notation.
For example, other books may say ms-2
instead of m/s2. As long as you use what
you’re comfortable with, you’ll be fine.
Why aren’t the units of frequency
written as cycles/s? Why just “1/s”?
A:
Good question. The thing to remember
is that frequency is the number (of things)
per second. It could be the number of
revolutions, or the number of waves that
break on the beach or the number of times
a dog barks. A number is dimensionless so
has no units.
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Then the frequency would be smaller
than 1. For example, something that
happens every 5 s has a frequency of 0.2 Hz.
1
1
Why does f = T and T = f ?
Frequency is cycles per second;
period is seconds per cycle. “Per” means
“divided by”, so frequency and period are the
“opposite way up” from each other.
For example, if something happens 5 times
per second, it has a frequency of 5 Hz.
Another way of putting it is that the thing
1
happens every 0.2 s because T = f = 0.2 s.