We're now going to learn a little bit about gravity.
And just so you know, gravity is something that, especially
in introductory physics or even reasonably advanced
physics, we can learn how to calculate it, we can learn how
to realize what are the important variables in it, but
it's something that's really not well understood.
Even once you learn general relativity, if you do get
there, I have to say, you can kind of say, oh, well, it's
the warping of space time and all of this, but it's hard to
get an intuition of why two objects, just because they
have this thing called mass, they are
attracted to each other.
It's really, at least to me, a little bit mystical.
But with that said, let's learn to deal with gravity.
And we'll do that learning Newton's Law of Gravity, and
this works for most purposes.
So Newton's Law of Gravity says that the force between
two masses, and that's the gravitational force, is equal
to the gravitational constant G times the mass of the first
object times the mass of the second object divided by the
distance between the two objects squared.
So that's simple enough.
So let's play around with this, and see if we can get
some results that look reasonably familiar to us.
So let's use this formula to figure out what the
acceleration, the gravitational acceleration, is
at the surface of the Earth.
So let's draw the Earth, just so we know what
we're talking about.
So that's my Earth.
And let's say we want to figure out the gravitational
acceleration on Sal.
That's me.
And so how do we apply this equation to figure out how
much I'm accelerating down towards the center of Earth or
the Earth's center of mass?
The force is equal to-- so what's this big G thing?
The G is the universal gravitational constant.
Although, as far as I know, and I'm not an expert on this,
I actually think its measurement can change.
It's not truly, truly a constant, or I guess when on
different scales, it can be a little bit different.
But for our purposes, it is a constant, and the constant in
most physics classes, is this: 6.67 times 10 to the negative
11th meters cubed per kilogram seconds squared.
I know these units are crazy, but all you have to realize is
these are just the units needed, that when you multiply
it times a mass and a mass divided by a distance squared,
you get Newtons, or kilogram meters per second squared.
So we won't worry so much about the units right now.
Just realize that you're going to have to work with meters in
kilograms seconds.
So let's just write that number down.
I'll change colors to keep it interesting.
6.67 times 10 to the negative 11th, and we want to know the
acceleration on Sal, so m1 is the mass of Sal.
And I don't feel like revealing my mass in this
video, so I'll just leave it as a variable.
And then what's the mass 2?
It's the mass of Earth.
And I wrote that here.
I looked it up on Wikipedia.
This is the mass of Earth.
So I multiply it times the mass of Earth, times 5.97
times 10 to the 24th kilograms-- weighs a little
bit, not weighs, is a little bit more massive than Sal--
divided by the distance squared.
Now, you might say, well, what's the distance between
someone standing on the Earth and the Earth?
Well, it's zero because they're touching the Earth.
But it's important to realize that the distance between the
two objects, especially when we're talking about the
universal law of gravitation, is the distance between their
center of masses.
For all general purposes, my center of mass, maybe it's
like three feet above the ground, because
I'm not that tall.
It's probably a little bit lower than that, actually.
Anyway, my center of mass might be three feet above the
ground, and where's Earth's center of mass?
Well, it's at the center of Earth, so we have to know the
radius of Earth, right?
So the radius of Earth is-- I also looked it up on
Wikipedia-- 6,371 kilometers.
How many meters is that?
It's 6 million meters, right?
And then, you know, the extra meter to get to my center of
mass, we can ignore for now, because it would be .001, so
we'll ignore that for now.
So it's 6-- and soon.
I'll write it in scientific notation since everything else
is in scientific notation-- 6.371 times 10 to the sixth
meters, right?
6,000 kilometers is 6 million meters.
So let's write that down.
So the distance is going to be 6.37 times 10
to the sixth meters.
We have to square that.
Remember, it's distance squared.
So let's see if we can simplify this a little bit.
Let's just multiply those top numbers first. Force is equal
to-- let's bring the variable out.
Mass of Sal times-- let's do this top part.
So we have 6.67 times 5.97 is equal to 39.82.
And I just multiplied this times this, so now I have to
multiply the 10's.
So 10 to the negative 11th times 10 to the negative 24th.
We can just add the exponents.
They have the same base.
So what's 24 minus 11?
It's 10 to the 13th, right?
And then what does the denominator look like?
It's going to be the 6.37 squared times 10
to the sixth squared.
So it's going to be-- whatever this is is going to be like 37
or something-- times-- what's 10 to the sixth squared?
It's 10 to the 12th, right?
10 to the 12th.
So let's figure out what 6.37 squared is.
This little calculator I have doesn't have squared, so I
have to-- so it's 40.58.
And so simplifying it, the force is equal to the mass of
Sal times-- let's divide, 39.82 divided by 40.58 is
equal to 9.81.
That's just this divided by this.
And then 10 to the 13th divided by 10 to the 12th.
Actually no, this isn't 9.81.
Sorry, it's 0.981.
0.981, and then 10 to the 13th divided by 10 to the 12th is
just 10, right?
10 to the first, times 10, so what's 0.981 times 10?
Well, the force is equal to 9.81 times the mass of Sal.
And where does this get us?
How can we figure out the acceleration right now?
Well, force is just mass times acceleration, right?
So that's also going to just be equal to the acceleration
of gravity-- that's supposed to be a small g there-- times
the mass of Sal, right?
So we know the gravitational force is 9.81 times the mass
of Sal, and we also know that that's the same thing as the
acceleration of gravity times the mass of Sal.
We can divide both sides by the mass of Sal, and we have
the acceleration of gravity.
And if we had used the units the whole way, you would have
seen that it is kilograms meters per second squared.
And we have just shown that, at least based on the numbers
that they've given in Wikipedia, the acceleration of
gravity on the surface of the Earth is almost exactly what
we've been using in all the projectile motion problems.
It's 9.8 meters per second squared.
That's exciting.
So let's do another quick problem with gravity, because
I've got two minutes.
Let's say there's another planet called the
planet Small Earth.
And let's say the radius of Small Earth is equal to 1/2
the radius of Earth and the mass of Small Earth is equal
to 1/2 the mass of Earth.
So what's the pull of gravity on any object, say same
object, on this?
How much smaller would it be on this planet?
Well, actually let me save that to the next video,
because I hate being rushed.
So I'll see you