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## Physics library

### Course: Physics library > Unit 4

Lesson 3: Newton's law of gravitation- Introduction to gravity
- Mass and weight clarification
- Gravity for astronauts in orbit
- Would a brick or feather fall faster?
- Acceleration due to gravity at the space station
- Space station speed in orbit
- Introduction to Newton's law of gravitation
- Gravitation (part 2)

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# Gravitation (part 2)

A little bit more on gravity. Created by Sal Khan.

## Want to join the conversation?

- well moon is smaller but its gravity isnt double much less more than earth's gravity.why?i dont understand.

also what is the comparison between the two the planets? is it of same density but more compressed to get half the radius?(21 votes)- Gravitational acceleration at the surface of a body is:

a = GM/R^2

assuming a uniform density ρ and spherical body, the mass will be

M = ρV = ρ * 4/3πR^3

substitute that M into the acceleration equation...

a = G/R^2 * ρ * 4/3πR^3

a = 4/3GρπR

so the gravitational acceleration at the surface increases linearly with the radius of the body.(12 votes)

- gravitational field is the same as gravitational acceleration? if no, why its symbol is g?

why is g=a ? accelaration is the variation of the velocity in a certain time, so the earth moves if someone, for example, falls from a building. please someont to explain, i am really confused(2 votes)- Surely you've noticed that when you "convert" a mass to a weight (on earth) you use 9.8 N/kg, and you also know that the acceleration due to gravity on earth is 9.8 m/s^2. Is it coincidence that they are both 9.8? Nope.

g stands for gravitational field strength AND acceleration due to gravity. They are the same thing. Let's calculate the acceleration of something if we drop it. The force on it is mg. Newton's second law tells us F = ma.

So

mg = ma

m cancels out

g = a

So the acceleration will be exactly equal to the gravitational field strength, and mass does not matter.

This is why everything falls with the same acceleration (if we can neglect air resistance)

Now let's look at the units.

Gravitational field strength has units of Newtons/kg. That's why the force on a 10 kg mass is 98 N, because (9.8 N/kg)*10 kg = 98 N.

The units of acceleration of course are m/s^2. So how can g be BOTH gravitation field strength AND acceleration due to gravity? Let's look more closely at the units:

A newton is a kg*m/s^2

gravitational field strength is in N/kg

So g = 9.8 N/kg = (9.8 kg*m/s^2)/kg = 9.8 m/s^2

In other words, N/kg is the same thing as m/s^2.(6 votes)

- Why we call G as universal gravitational constant if it is derived by taking only earth's mass and acceleration due to gravity on earth 'g'.and distance between center of masses of masses of earth and object.If we have taken everything of earth here then why it is universal?

as we know Fg=(Gm1.m2)/r^2

then we find out G's value as G=(Fg.r^2)/(m1.m2).

here Fg is gravitational force on earth,r is distance between object having mass m1 on earth's surface and center of mass of earth having mass m2.(3 votes)- G is a constant measured by Henry Cavendish. It is universal, but g is not. We did not calculate the value of G with the value of g. It was the other way round(3 votes)

- i didn't get the concept of higher gravitational attraction of a smaller planet than earths(ok,if its the radius adjusting itself with the masses),then how come moons gravity less than that of earths?(1 vote)
- It's not about size but about mass. In short: The heavier the object, the greater its gravity. Black holes, for example, are reeeally small (compared to the rest of the universe) but incredibly dense and thus incredibly heavy. This is what causes them to attract everything that comes too close to it - even light - and never let it go. So, if there's a planet that is smaller but heavier than Earth, its gravitational force will be stronger. And the moon's gravity is less than the Earth's because it is much lighter, as Andrew already said.(4 votes)

- If. The earth suddenly stopped in its orbit assumed to be circular how long it might take before it falls into the sun(1 vote)
- so.....now let's say a body is moving with uniform acceleration in space.......

this means that since there is no friction in space it keeps accelerating and eventually reach the speed of light and probably travel faster that light........

so.......can there be objects that are faster than light?(2 votes)- No, because Newtonian physics does not apply as you get close to the speed of light.

The object's mass increases, so it becomes harder to accelerate as it gets close to light speed

The mass approaches infinity as speed approaches c

Nothing with mass can travel at or above the speed of light(3 votes)

- If G is there to cancel the units, why int it then just a whole number like 1?(1 vote)
- To cancel units is one purpose of the constant G is is also there to adjust the value so that the units produce the correct value in the same way that if you get an answer of length in meters and you want it in centimeters you multiply a by 100.(4 votes)

- If a smaller planet has a larger gravitation, how does the moon has a smaller gravitational force than Earth?(2 votes)
- You've misunderstood, smaller planets don't have larger g, unless they have the same mass as larger planets.

g at the surface depends on both mass and radius. The moon's mass is much less than Earth's.(2 votes)

- The distance between two electrons in an orbit ( I am using that concept just for this example) is very small, so the gravitational force between them should become immeasurably strong. So how come they are still repelled by the electromagnetic force? Shouldn't the gravitational force between them be stronger because of the very small distance, or at least be able to balance the Coulomb force?(1 vote)
- Nope. The distance is used for calculation both the gravitational and electromagnetic force.

The electromagnetic force is greater than the gravitational force (so much that you don't even consider the gravitational one :D) because the electron has a huge charge, compared to it's mass.

It's mass is as big as this: https://www.google.com/search?q=electron+mass which is... not that heavy :D(4 votes)

- Is there any difference in the magnitude of inertial and gravitational mass?(2 votes)
- No, as far as we can tell they are exactly equal. We have measured this to about 13 decimal places, if I recall correctly. Einstein predicts that they are exactly the same.(2 votes)

## Video transcript

Welcome back. So I was trying to rush and
finish a problem in the last two minutes of the video, and
I realize that's just bad teaching, because I
end up rushing. So this is the problem we were
going to work on, and you'll see a lot of these. They just want you to become
familiar with the variables in Newton's law of gravitation. So I said that there's two
planets, one is Earth. Now I have time to draw things,
so that's Earth. And then there's Small Earth. And Small Earth-- well, maybe
I'll just call it the small planet, so we don't
get confused. It's green, showing that
there's probably life on that planet. Let's say it has 1/2 the radius,
and 1/2 the mass. So if you think about
it, it's probably a lot denser than Earth. That's a good problem
to think about. How much denser is it, right? Because if you have 1/2 the
radius, your volume is much less than 1/2. I don't want to go into that
now, but that's something for you to think about. But my question is what
fraction, if I'm standing on the surface of this-- so the
same person, so Sal, if I'm on Earth, what fraction is the pull
when I'm on this small green planet? So what is the pull
on me on Earth? Well, it's just going to be-- my
weight on Earth, the force on Earth, is going to be equal
to the gravitational constant times my mass, mass of me. So m sub m times the mass of
Earth divided by what? We learned in the last video,
divided by the distance between me and the center
of the mass of Earth. Really, my center of mass and
the center of mass of Earth. But this is between the surface
of the Earth, and I'd like to think that I'm not
short, but it's negligible between my center of mass and
the surface, so we'll just consider the radius
of the Earth. So we divide it by the radius
of the Earth squared. Using these same variables,
what's going to be the force on this other planet? So the force on the other
planet, this green planet-- I'll do it in green-- and we're
calling it the small planet, it equals what? It equals the gravitational
constant again. And my mass doesn't change when
I go from one planet to another, right? Its mass now is what? We would write it m
sub s here, right? This is the small planet. And we wrote right here that
it's 1/2 the mass of Earth, so I'll just write that. So it's 1/2 the mass of Earth. And what's its radius? What's the radius now? I could just write the radius
of the small planet squared, but I'll say, well, we know. It's 1/2 the radius of Earth,
so let's put that in there. So 1/2 radius of Earth. We have to square it. Let's see what this
simplifies to. This equals-- so we can take
this 1/2 here-- 1/2G mass of me times mass of Earth over--
what's 1/2 squared? It's 1/4. Over 1/4 radius of
Earth squared. And what's 1/2 divided by 1/4? 1/4 goes into 1/2 two
times, right? Or another way you can think
about it is if you have a fraction in the denominator,
when you put it in the numerator, you flip it
and it becomes 4. So 4 times 1/2 is 2. Either way, it's just math. So the force on the small planet
is going to be equal to 1/2 divided by 1/4 is 2 times
G, mass of me, times mass of Earth, divided by the radius
of Earth squared. And if we look up here,
this is the same thing as this, right? It's identical. So we know that the force that
applied to me when I'm on the surface of the small planet is
actually two times the force applied on Earth, when
I go to Earth. And that's something interesting
to think about, because you might have said
initially, wow, you know, the mass of the object matters
a lot in gravity. The more massive the object,
the more it's going to pull on me. But what we see here is
that actually, no. When I'm on the surface of
this smaller planet, it's pulling even harder on me. And why is that? Well, because I'm actually
closer to its center of mass. And as we talked about earlier
in this video, this object is probably a lot denser. You could say it's only 1/2 the
mass, but it's much less than 1/2 of the volume, right? Because the volume is the cube
of the radius and all of that. I don't want to confuse you, but
this is just something to think about. So not only does the mass
matter, but the radius matters a lot. And the radius is actually the
square, so it actually matters even more. So that's something
that's pretty interesting to think about. And these are actually very
common problems when they just want to tell you, oh, you go to
a planet that is two times the mass of another planet, et
cetera, et cetera, what is the difference in force
between the two? And one thing I want you to
realize, actually, before I finish this video since I do
have some extra time, when we think about gravity, especially
with planets and all of that, you always
feel like, oh, it's Earth pulling on me. Let's say that this is the
Earth, and the Earth is huge, and this is a tiny spaceship
right here. It's traveling. You always think that
Earth is pulling on the spaceship, right? The gravitational
force of Earth. But it actually turns out, when
we looked at the formula, the formula is symmetric. It's not really saying one
is pulling on the other. They're actually saying
this is the force between the two objects. They're attracted
to each other. So if the Earth is pulling on
me with the force of 500 Newtons, it actually turns out
that I am pulling on the Earth with an equal and opposite
force of 5 Newtons. We're pulling towards
each other. It just feels like the Earth is,
at least from my point of view, that the Earth
is pulling to me. And we're actually both being
pulled towards the combined center of mass. So in this situation, let's say
the Earth is pulling on the spaceship with the force
of-- I don't know. I'm making up numbers
now, but let's say it's 1 million Newtons. It actually turns out that the
spaceship will be pulling on the Earth with the same force
of 1 million Newtons. And they're both going to be
moved to the combined system's center of mass. And the combined system's center
of mass since the Earth is so much more massive is
going to be very close to Earth's center of mass. It's probably going to
be very close to Earth's center of mass. It's going to be like
right there, right? So in this situation, Earth
won't be doing a lot of moving, but it will be pulled
in the direction of the spaceship, and the spaceship
will try to go to Earth's center of mass, but at some
point, probably the atmosphere, or the rock that it
runs into, it won't be able to go much further and
it might crash right around there. Anyway, I wanted just to give
you the sense that it's not necessarily one object just
pulling on the other. They're pulling towards
each other to their combined center of masses. It would make a lot more sense
if they had just two people floating in space, they actually
would have some gravity towards each other. It's almost a little romantic. They would float
to each other. And actually, you could
figure it out. I don't have the time to do
it, but you could use this formula and use the constant,
and you could figure out, well, what is the gravitational
attraction between two people? And what you'll see is that
between two people floating in space, there are other forms
of attraction that are probably stronger than their gravitational attraction, anyway. I'll let you ponder that
and I will see you in the next video.