This lecture is going to be about uniform circular motion, which is what it
sounds like: an object moving in a circle. Let's say that you want to make an
object move in a circle. What would you need to do to it to make that happen?
Well you can't just let it continue with what it's doing without applying some
force to it, because if you do that, it's just going to continue with whatever
velocity it already had according to Newton's First Law. So let's say that you
have this cart moving toward you with some velocity and you want to make it
move around you, and I'm going to imagine that you have this rope that you extend
out to the cart to make the cart move in a circle as it goes by you.
You'd have to apply a constant net force toward the center of the circle that
would cause the cart to move in a circle. So this is based on Newton's First Law.
An object maintains its velocity unless acted on by a force, so if that force
from you disappears at any point the object is just going to continue off
with whatever velocity it had in the moment that you let go in whatever
direction that it had in the moment that you let go. And let's say that the object
had no initial velocity. If it didn't have any initial velocity to start off
and you apply this force to it, it would just be dragged toward you without any
circular motion. It's just moving toward you. So this shows that there are two
requirements for circular motion: the first requirement is that the net force
has to be constant and always pointing toward the center of the circular path,
and the second requirement is that there has to be a constant velocity at a 90
degree angle to the net force itself, so this picture on the left kind of
shows you what I mean by that. So both of these things have to be true for
circular motion to occur. This gives us some vocabulary that we can use to talk
about circular motion. The centripetal force is a name that we give to the net
force on the object that's moving in a circle. It always points toward the
centre of motion, and it's measured in Newtons.
the tangential velocity is the distance the object moves around the circle over
the change in time. It's measured in meters per second. The radius of the
circle is just the radius of the circular path, which is measured in
meters, and the centripetal acceleration means the acceleration that points
toward the center. Because acceleration always points in the same direction as
the net force, that means that in circular motion the acceleration is
always going to be pointing toward the center of the circle. So this picture
shows you what I mean and how these variables are related over time. So you
can see that as the circle goes around the velocity is
always pointing tangent to the circular path (parallel to the circular path) and
the net force and acceleration are always pointing toward the center of the
path. A common misconception about centripetal force is: students often
think that centripetal force is a type of force, like there's centripetal force
and there's the force of gravity, there's the normal force, but that's not correct.
Centripetal force is just the net total force that always points toward the
center of motion, and any force can add together to make it. So as an example:
when earth is orbiting around the sun gravity is the centripetal force, so the
centripetal force is just a name that we give to whatever force happens to be
pointing toward the center constantly. There's no such thing as a type of force
called "centripetal force" that's always made up of other types of forces, so some
students are also confused about why the acceleration always points toward the
center of the circle, whereas the velocity is always pointing tangent to
the circle. One way to think about this is that the net force points toward the
center and acceleration always points in the same direction as the net force. A
second way is to watch the vector animation that's happening above my head.
You can see that because acceleration is the change in velocity, it's causing
velocity to always change toward the center of the circle. So the velocity
continues to turn around, and that's why circular motion is maintained.
If acceleration pointed in the same direction as velocity the object would
just continue along a straight line path. Another common misconception about
centripetal acceleration is that it means that the object is getting faster
and faster around the circle, but that's also not true. The role of the
centripetal acceleration is to change the direction of the velocity, but it
doesn't change the magnitude of the velocity, just the direction, so the
velocity is staying at a constant rate. It''s not getting bigger or smaller. The
acceleration is just changing its direction as it turns around the circle.
Two more vocabulary words that you'll need to describe circular motion are
period and frequency. Period is the amount of time an object takes to
complete one full cycle. We use the word period to describe a lot of different
things. For example, class periods are cycles that repeat over a given amount
of time, and just like that a period in physics is just the amount of time that
any cycle in physics takes to repeat. Here in circular motion specifically it
means how long the object takes to go around the circle once, and it's always
measured in seconds, and the unit for it is capital T. Frequency is the amount of
repetitions a cycle completes in one second, so here in circular motion
frequency means how many times the object goes around the circle in one
second. Frequency is measured in Hertz or HZ. HZ just means events per second, so
period is how long it takes for an event to repeat once, and frequency is how many
events repeat in one second. So there's an interesting relationship there where
the period is always going to be equal to one over the frequency, this is always
going to be true. I'm gonna give you a visual that's gonna help you understand
why. Let's say that you're going around a circular path and it takes you four
seconds to go around the path altogether, because it takes you four seconds to
repeat this cycle once. That's your period, the period is four seconds long
because that's the time it takes to repeat the frequency is the amount of
repetitions completed in one second, and if you look at my visual you can see
that in just one second one fourth of the total circle has been covered. So
that means that the circle is being covered at 1/4th rotations per
second. So the frequency here is going to be 1/4th, and you can see from this
that no matter what fraction I choose the period is always going to be equal
to one over the frequency, so here the period is 4 and the frequency is 1/4th.
and that is always going to be true. So if you have the period of some
circular motion you also have the frequency. One more important vocab word
is angular velocity. I'm going to compare that to tangential velocity with this
animation. In this animation you can see I have a car driving in circular motion,
and next to the car's path you can see I put a boy sitting on some playground
equipment that also spins in a circle. So when I make these two things start to
spin you can see that they're both spinning around the circle, but the car
is covering a lot more distance over the same amount of time, so that means that
the car has a much larger velocity than the boy. If you look down at the car'S
speedometer and looked at how fast the boy was going you would see that the car
is going much faster than the boy. So that's tangential velocity, that's how
fast it's actually moving in meters per second. So you can see that the velocity
of the car is much greater than the velocity of the boy, but there's another
part of circular motion that we can look at where these two things are actually
equal to each other. So the angular velocity is just the change in angle
around a circle over the change in time. It's measured in radians per second, so
it's a measurement of how many radians around a circle you go per second. So
even though the boy's circular path is much smaller than the car's circular path,
if you watch how many degrees or radians they go around the circle in a given
amount of time you can see that it's actually the same. They're covering the
same angle of their respective circles in the same amount of time. That's useful
information, so we record that in physics as a separate variable: angular velocity.
That W symbol is actually an omega, so when you're drawing that you just need
to make sure the two sides curl in like that.
So you can see here that the angular velocity of the car, how much of an angle
it covers over time, is actually equal to the angular velocity of the boy, even
though the velocity, the actual tangential velocity of the car, is much
greater than the velocity of the boy. The car is much faster but it goes around
the circle at the same rate as the boy goes around his circle, so the angular
velocity is the same but the tangential velocity is different. So that was it for
definitions for the first part of this unit. I'm going to show you the equations
for this unit and where we get them. I'm gonna start with an equation for the
tangential velocity around the circle. We know that in general, velocity is
displacement over time. Here the object is going all the way around the circle
once, and if the circle has a radius R we know that the total distance that this
object covered is equal to 2 pi R. That's the circumference of the circle. And
based on our previous definition, we know that the time that it takes to complete
one full circle is equal to the period of motion. The period is the amount of
time in seconds that the object takes to rotate around the full circle, so our
first equation that we're going to be using a lot is that the tangential
velocity of any object is equal to 2 pi R over the period of motion like that,
and because period is 1 over frequency we can rewrite that equation as
tangential velocity is equal to 2 Pi R times the frequency. So that's the first
equation. We're going to use the second equation for angular velocity. We know
that angular velocity is the change in degree over time or radians per second,
and I know that if this object goes all the way around the circle it covers two
pi radians in that time, and the time that it takes to do that is the period,
so that's going to be 2 pi over T is the angular velocity, and because T is 1 over
F I can rewrite that as 2 pi F. So that's an equation for angular velocity, and I'm
going to keep this period and frequency equation up here as well.
The next thing you might notice is that the tangential velocity is actually
equal to the angular velocity multiplied by the radius. Each equation for the
tangential velocity is actually equal to each equation for the angular velocity
multiplied by the radius, so this equation is going to hold true. This is
an equation for the centripetal acceleration of the object: how much it's
accelerating toward the center of its circular path.
I'm not gonna prove that in this video. It's a little complicated, but if you're
interested in knowing where it comes from I've left a proof in the
description of the video so centripetal acceleration acceleration toward the
centre of motion is equal to the tangential velocity squared over R and
the centripetal force on an object is the net force so just like in regular
straight-line physics the net force is equal to M times a here the centripetal
force is equal to mass times the centripetal acceleration, which is also
equal to mass times V squared over R. So in conclusion this is all the new
vocabulary for the unit. There's a lot, but once you get used to using it each
individual idea is pretty straight- forward. I'm gonna end the video with two
examples of circular motion problems and what they look like. A thousand kilogram
car takes 4 seconds to turn from north to west around a corner. If the
radius of curvature of the road is thirty meters, what force of friction do
the car's wheels apply? So we can imagine this car going around the circle and the
force that's taking the place of the centripetal force here is gonna be the
force of friction. This is because it has to be some force pointing toward the
center of the car's circular path. So that has to be some horizontal force, and the
only horizontal force that can exist here is some push from the road itself,
which kind of makes sense because if the road is slippery, if there's ice, it's
much more difficult to turn. Your car just keeps going with whatever velocity
it already has. So the force that's acting as the centripetal force here is
the force of friction. So we know that that is what we're trying to solve for.
This says that the radius of curvature of the road is 30 meters, which means
that if this curve continued to curve into a complete circle, that circle would
have a radius of 30, so that's going to be important for our calculations. The
mass of the car is a thousand kilograms and because it takes four seconds to
turn from north to west I know that that's going to be one-quarter, that turn
is one-quarter of the total circle that you're seeing. So if that takes four
seconds to turn, all together the total circle would take 16 seconds for the car
to turn around. So the period of motion here is going to be 16 seconds. So I have
the radius, the mass, the period, and I need to find the centripetal force. So
looking at my centripetal force equation, I'm probably going to need to solve for
what Vt is because I can plug in my mass, I can plug in my radius, and I know that
Vt by itself is equal to 2 pi r over capital T. So plugging that stuff in I
get 11.8 meters per second. Plugging that into the original equation
I get that the centripetal force on the car is four thousand six hundred
forty-one Newtons. A lot of the unit is going to be like this where you're
picking out individual pieces of information and plugging it into
equations. There are some parts that are very conceptual but a lot of it is just
learning to recognize which variable is what and plugging in the information
correctly. Here's example 2: The distance from the earth to the Sun is
149.6 million kilometers. How fast is the earth moving
around its orbit? what is the Earth's acceleration toward
the sun? So this is an example of a problem that gives us some context clues
for solving. We're trying to find the tangential velocity and the centripetal
acceleration of the earth ,and we're given the radius of Earth's circular
path because if the distance from the earth to the Sun is that far and the sun
is in the center of the Earth's orbit that's also going to be the radius of
the Earth's orbit. This is going to come up a lot, so if I convert that number to
scientific notation and convert it to meters I get 1.496
times 10 to the 11th meters for the radius of that circular path. So to solve
for the tangential velocity I need the radius and the period, and the problem
doesn't give me the period but I actually know that using the context of
the problem because the period is how long in seconds the object takes to go
around the circular path once, and I know that it takes the earth one year to
complete its circular path. That's what a year is, so a yea is 365 days so we just
need to convert that to seconds. And when I do that I get that the period is three
point one five times ten to the seventh seconds. Plugging that into two PI R over
T I get two point nine eight times ten to the fifth meters per second, and the
centripetal acceleration is just V squared over R. So just plugging in my
numbers here I get this very small acceleration for Earth. It makes sense
that it's accelerating at such a small rate because it's only changing its
direction at a very small rate, because it takes a full year to completely
change its velocity's direction all the way around the circle. So that's how you
solve circular motion problems. A lot of it is just going to be drawing out those
variables, plugging them into the equations, and finding missing
information.