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.