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## High school physics - NGSS

### Course: High school physics - NGSS>Unit 2

Lesson 1: Newton's law of universal gravitation

# Introduction to gravity

Newton's Law of Universal Gravitation describes the strength of gravitational attraction between two objects. The gravitational force is equal to the mass of object 1 times the mass of object 2, divided by the distance between the objects squared, all times the gravitational constant (G). G is small, so the force is only noticeable if one or both objects are very massive. Created by Sal Khan.

## Want to join the conversation?

• how can birds resist gravity and fly?
(182 votes)
• It is the same principle as an airplane that is taking off. It has the thrust and the lift and so can resist gravity. In the case of the bird the lift comes from all of the individual feathers and the thrust comes from the bird flapping its wings or finding a hot air current to find lift on. That is another story though.
(54 votes)
• Where does gravity ultimately come from? Do masses naturally have gravitational forces associated with them? Since you can convert Mass into Energy (E=MC^2), why doesn't energy, such as light, have gravity? I've read some where that gravity is theorized to be the resultant force of the big bang. It is not objects are attracted to each other, but everything in the universe is trying to move back toward the origin of the universe. Kind of like Newton's 3rd law. Please explain :D
(94 votes)
• @terryjbrennan You are incorrect. Photons are, by nature, massless. That is why they are able to travel at the speed of light (C) and not just below it. Anything with mass cannot move at the speed of light.

@kokocipher In answer to your question "where does gravity ultimately come from," according to the General Theory of Relativity, the most widely accepted and most accurate theory used to explain our observations of gravity, gravity is the result of objects with mass bending space time. All objects bend spacetime, and more massive/dense objects bend it more steeply. That is they have a stronger gravitational pull. Since mass == energy (E=MC2)energy will indeed bend space time, if there is enough energy in a small enough space.

However, consider the following. It requires the mass of an entire moon or planet to have an effect strong enough for you to feel. Since E=MC2 the amount of energy required to bend space the same amount (that is to have a gravitational influence that humans could detect without instruments) is unfathomable. If there's energy that dense it tends to collapse into matter. Indeed, this is what happens in our particle accelerators such as the LHC.

As for black holes, they are locations in space where there is mass dense enough to bend spacetime so steeply that it would require movement faster than the speed of light to escape. Due to the nature of our Universe it is impossible to exceed the speed of light (I'm aware that sounds vague, but it would take a good hour to explain in more detail). Therefore, nothing, not even light, can escape.
(119 votes)
• Where does the Big G come from and why is it that specific number?
(38 votes)
• The constant G can be calculated from data from experiments. This was first done by Cavending 111 years after Newton published the formula. Basically, he measured the gravitational force between a couple of lead balls by seeing how much they twisted a metal wire. Read the details of this experiment at http://en.wikipedia.org/wiki/Cavendish_experiment.
(44 votes)
• How come a helium balloon resist gravity ?
(27 votes)
• helium has very less density so it apparently floats in air because air has a greater density. just as how ice floats on water because ice as a lower density than water. the same way helium has less density than air so it floats and does not defy gravity.
(45 votes)
• Gasses have very high kinetic energy. DO they escape earth's gravitational pull?
(13 votes)
• On very rare occasions. A gas molecule, like any other escaping projectile, requires enough velocity (escape velocity) to leave from an orbiting path around the earth. I suppose if a gas molecule in the atmosphere received enough energy (and turned to a near plasma state) it would be possible.
One issue is that while individual gas molecules have high kinetic energies, they are characterized by randomly colliding with other gas molecules at high speeds. So you wouldn't really see any synchronized movement of molecules away from the earth's surface--while some are moving away, others are moving towards the earth, and they collide with each other.
(22 votes)
• How is the mass of earth calculated?
(6 votes)
• This was basically a three step process. 1) First of all, you have Newton's Law of Universal Gravitation, that states that the force of attraction due to gravity between two masses, m and M, at a distance r, is given by F=GmM/r^2, where G is a constant called the gravitational constant. 2) Second, you know that all objects on the Earth's surface will fall with a constant acceleration, known as g. From Newton's second law of motion, F=ma, you get that mg=GmM/R^2, where M is the Earth's mass and R is the Earth's radius. Do a little algebra, and you get that M=gR^2/G. g is measured, and so is R. All you need is G (and love). 3) Enters Cavendish, who measured the gravitational attraction between two spheres of known mass and distance, therefore finding G. You plug in the values, and you get M. Awesome!
(18 votes)
• I am a little confused at this point so my question might be invalid but lemme give it a try :
An apply thrown in the air also attracts the earth but due to the mass of the earth the acceleration on earth is negligible , but then if a huge amount of apples were thrown from a single point on earth then would the earth move towards the apples due to the combined force of all the apples thrown in the sky??
(6 votes)
• Absolutely correct, but you will need millions if not billions, trillions, or more apples
(9 votes)
• Hello,

This might be slightly off-topic, but I had a doubt I couldn't clarify.

Say, two objects of masses M and m are placed at rest in vacuum, a distance R away from each other and the only forces acting upon them are each others' gravitational forces. If they start accelerating towards each other, how long would it take for them to collide?

I tried solving it through integration and failed. (Equating initial P.E. and final K.E. didn't work well (for me) either.) The best expression for time I could come up with was,
``t = ____√(2R^3)____      3∙√(G(M+m))``
Since this is a made up question, I have no clue whether it is correct or not (probably isn't), so I would appreciate it if someone could help out.

[Edit: Through a bit of trial and error, I think I got the right answer as
``t = ____√(2R^3)____,      √(3∙G(M+m))``
though I am not sure how to derive it.

Note: I replace the numerator with `√(2((Re+h)^3 - Re^3))`, where Re is the radius of the earth
h is the height above the surface of the earth,
M and m are the masses of the earth and another object.

I compared the result with the result from `h = ut + 1/2(at^2)` , and got an answer within 4 decimal places (for h << R). Still not sure how I got the √3 in the denominator, though.]

Thank you
(8 votes)
• Let's take a different approach.

Imagine the two masses are in orbit with mass M and m.

Orbit is just continous freefall, which is why the ISS experiences no gravity (or microgravity).

We make the orbits very very eecentric, long and skinny so the semi minor axis is 0, leading to a straight line across the semi major axis.

The bigger mass, M, has a higher F_g but in return, it has a higher inertia. The lower mass, m, has a lower F_g but has a lower inertia and moves quite easily towards the big mass.

Like a pendulum, these two forces cancel equally, leaving the distance 1/2 the distance between them, or in other words R/2.

These are the objects: O ---> F_g * F_g <- o

So they will meet at the center, R/2.

But remember, these are just regular circular orbits. But, the mass of the smaller object has enough mass to shift the center of mass of the system, in other words bring the barycenter out of the main object.

The semi major axis of these objects are just half the distance between their circular orbits, ignoring the barycenter.

So its: a_1 + a_2: a_2 + a_1 for the big mass and small mass, respectively.

But then we need to include the barycenter, so we need to take the average of those semi-major axes because the barycenter is exactly half the distance between those objects.

So: (2a_1 + 2a_2)/2 = a_1 + a_2. Since before we made these orbits very long and skinny, these were ciruclar orbits so a_1 and a_2 are the same distance, 1/4 of the total distance.

The orbit equation for their time which is part of
Kepler's Laws is: T = 2pi*sqrt(a^3/GM)

But we need to remind ourselves that this is a 2 body problem.

If we look at F = ma, this is only for 1 object.

For 2 objects, we need to include both so the F = ma equation turns out to be this:

summation(F) = summation(m)a.

So we need to add the two forces of the objects and the two masses of the objects to get the total acceleration.

In this case, we want the total time, so we need to apply the same logic.

a is semi major axis by the way.

So: T = 2pi*sqrt[summartion(a)^3/G*summation(M)]

So: T = 2pi*sqrt[(1/4R+ 1/4R)^3)/G(M + m)]

Which is: 2pi*sqrt[(R/2)^3/G(M+m)]

Thus: 2pi*sqrt[(R^2/8)/G(M+m)]

...a complex fraction where if we simplify it:

T = 2pi*sqrt[R^3/8G(M+m)]

But we want half the time for the collision, so divide that by 2:

T = pi*sqrt[R^3/8G(M+m)]

You could do further simplyfing

If you would like to get better details, it is explained well in this video: youtube.com/watch?v=F0zUOcZN6sI

Hope this helps!
(5 votes)
• If photons are massless particles, then why does gravity bend light?
(8 votes)
• Photons have no rest mass, but they do have energy, and from relativity energy and mass are equivalent, so the photon will interact with gravity for that reason. The other way of looking at it is what was mentioned before, that gravity curves spacetime and light follows the curved spacetime path.
(5 votes)
• I'm gonna go a step further than Newton :D and ask why do things have to follow the Universal Law of Gravitation? I mean, like, what causes this gravitational force? For e.g., electrostatic force is created due to charges... then gravitational force is caused due to what??
(5 votes)
• Why does it need a cause? It could simply be the way it is.

Electromagnetism isn't created by charges, rather a charge is simply how a particle interacts with electromagnetic fields, using photons as force particles(aka bosons). In a similar way, it is thought particles with mass or energy interact with the gravitational field via bosons called gravitons.
(5 votes)

## Video transcript

Everything in human experience, and really human history or human civilizations experience, is that everything seems to fall to the earth, that if you have water particles, they don't just float up. If they are small enough they are being held up by the wind and all that, but if they are large enough, they will fall. That you don't have people that are able to just float around, they will fall. You don't have taxi cabs that float around, they'll fall. Not only will the water fall, it will hit the ground, it will puddle up, and if there is a gutter it will fall into the gutter. It is just trying to get lower and lower and lower. If i were to drop a bunch of needles they would just fall. They don't... If i had a needle at rest here it doesn't just automatically for no reason jump and fly upwards and start to float. And so it is just a thing that is fundamental to everything that we have ever ever experienced. And so, for most of human history or human civilization, we just accepted it as a given. We thought, "well look it's just obvious, everything should just fall down, that's just the way the universe is. To think otherwise would just be crazy." And that's why this guy, this guy right over here, is one of the greatest geniuses of all time. He did many more things than just the things I am going to describe in this video, and any one of those things would have earned him his place in history. And this, as you may already know, is Isaac Newton. Easily one of the top five minds in all of human history. So a pretty fascinating dude. And one of his big insights about this 'things falling down' problem is: Do they have to fall down? Is this just something we should assume about the universe? Things just need to fall down, he said, or we were told he said that he was somewhat inspired by observing an apple falling from a tree. It's probably not true that you'll see in some cartoons on television that the apple hit his head or hit his head while he was sleeping and gave him the idea. Most people were to see... Let me draw a tree here. That's a twig right there, some leaves. So if most people... So that is an apple over here. So if most people... If i were to snap this twig over here, the apple would fall. Pretty much common sense. The apple would fall. And if most people were to see that, they would just think it's a normal happening in the universe. But for Isaac Newton, at least on that day, he asked himself, "Why? Why did that apple fall?" And this, to some degree, is a great example of "out of the box" thinking, because something for thousands of years or tens of thousands of years human beings had taken for granted, just because that's the way it always was. He actually asked the question why? Does it always have to be that way? And that question took him down a entire line of reasoning that set up the basis for all of classical mechanics for the most part we still use today. It has been tweaked a good bit by this gentleman in the last hundred years. [Albert Einstein] But for most purposes, when we're engineering things on the surface of the planet, and we are not going close to the speed of light, we can still use the mathematics that Isaac Newton came up with from this simple question. And not only was he able to kind of think that there's something... There's something that might be pulling, somehow, acting on this apple, bringing it to the earth. But he actually formulated an entire, an entire.. I guess Law around this thing. So, as you can imagine, the thing that Isaac Newton believes brought the apple to the Earth is gravity... is gravity. And he formulated the universal Law of gravitation, or the law of universal gravitation; either way. And in it, he theorizes that the forces between objects now it's a vector quantity, it's always going to attract the two objects to each other. So the direction is towards each other. The force of gravity between two objects... is going to be equal to this this big G, which is really just a number, its a very small number. I'll give you that number in a second. It's equal to this constant, this gravitation constant. Which is a super-duper small number, times the mass of the first object, times the mass of the second object, divided by the distance between the two objects. Distance squared. So this is distance between two objects. So if you're talking about the force of gravity on Earth, this right over here... You pick one of the masses to be Earth, so this mass over here. You pick one to be Earth. This is the object on Earth. Maybe it's me. Then this is the distance between the center of masses between those two objects. The center of me and center of the Earth. So really it's from roughly the distance from the surface of the Earth or if I'm roughly five foot-nine [inches tall] then about half of that distance to the center of the Earth is this number right over here. So right when you see this, before we even talk about me and Earth, or needles and Earth, or taxi cabs and Earth and that force of gravity, you might have something bizarre raising up in your brain. You might be saying, "the way gravity is defined by Isaac Newton, this formula we're given right here, it's saying we have gravity between any two objects" and you're saying, "Look, I'm looking at a computer screen right now, so you're looking at a computer screen right now..." And let me draw an old school computer, not a flat panel. How come your not attracted to the computer screen? How come it doesn't fly into your face? And the answer there is, this number, this number is really small. And there actually is some force of attraction between you and the computer. It's just that it's more than overcompensated for the friction between the computer and the desk, the friction between you and your seat, which is frankly being caused by the force between you and the Earth, the force of gravitation between the computer and the Earth. That you and the computer have such small masses that you really can't notice it. It's really negligible. It's being overpowered by other forces that are keeping the computer from even drifting into your face or your face drifting into the computer. So just to get a sense of that... This G, this big G, this constant of proportionality, just to get a sense of how small it is... This is, and I'm going to round it here, it's approximately 6.67 times 10 to the negative 11th Newtons. And we'll talk about Newtons, it is the metric unit of force. Let me actually make sure I say this correctly. Newton meters per kilogram squared. Newton times meter per kilogram, squared. It's this strange set of units here, but the units are really there. So when you multiply by two masses, which are in kilograms, and divide by a distance, which is in meters, you'll end up with Newtons. But I want to make it clear that this is a super small number. Ten to the negative 11th. If I were to just write 10 to the negative 11th, it would be 0.0 and then we would have eleven 0's. So this number right here is the same as 6.67 times this thing over here. So this is a super small number. And that is why if you multipy it by not so large numbers, if you don't use Earth, if you use yourself and a computer, you're going to end up still with a super duper small force. Something so small that you won't notice it. It is going to be overpowered by other forces, so these things don't fly into each other. But when you think about really massive bodies, like the Earth, the force of gravity starts to become noticeable, very noticeable. And I'm not going to give you the mass of Earth in this video, you can look it up yourself. But if you put in the mass of Earth right over here, if you put it in right over there, and if you put in roughly the distance from the surface of the Earth to the center of the Earth for R, and you multiply that by G. All of these terms over here... So this term, if you multiply that times that term and multiply by this term squared, they simplify to what is sometimes called little g. Little g. So this right here, we can view that as the gravitational field at the surface of Earth. It's also the same thing as the acceleration of gravity at the surface of the Earth, and this, and once again I'm just going to round it for the sake... This is... This comes out to be, units wise, 9.8 meters per second squared. Then you're left with just the other mass. So times M1. So for simplicity, if something is close to the surface of the Earth, the distance does matter. We can say that the force of gravity can be this "little g" times whatever the mass is close to Earth. For example, if you were to take me, and I weighed 70 kilograms... So, in the case of Sal, Sal has a...Actually I shouldn't say weight, I have a mass of 70 kilograms. I have a mass of 70 kilograms. Weight is actually a force, but we'll talk about that, clarify that more later. My mass is 70 kilograms, then we can figure out the force that the Earth is pulling down on me which is actually my weight. So in this situation, the force is going to be g, which is 9.8 meters per second squared, times my mass, which is 70 kilograms. And let me get my handy T85 calculator out to figure this out. So I get 9.8 times 70. That gives me...686. So this is equal to 686 kilogram meters per second squared. OR this is the exact same thing as this thing right over here. This IS the definition of a Newton. So this is a newton, which is appropriately named for the guy that is the Father of All Classical Physics. So my weight on Earth, which is the same thing as the force that Earth is pulling down on me, or that the gravitational attraction between the Earth and me is 686 newtons. Now I will ask you an interesting question. So here is Earth, and I am not even a speck of a speck on Earth, but say for simplicity let's say this is me, I'm hanging out in the Indian Ocean some place. So that is me. And we already know that Earth is pulling down on me with a force of 686 newtons. Now my question to you is, "Am I pulling on the Earth with any force? And am I pulling on the Earth with a larger or small force that is pulling on me?" And your gut or your knee-jerk reaction might be: Earth is so huge, Sal is so tiny. Clearly the Earth must be pulling with a greater force than Sal is pulling on the Earth. Unfortunately that is NOT the case. That I am. So the Earth is pulling on Sal with the force of 686 Newtons and Sal is also pulling on the Earth with a force of 686 Newtons. So Sal is also pulling on the Earth. It makes me feel very powerful with the force of 686 Newtons. But you might say, "Wait, that doesn't make sense, Sal." If I have a building over here, and if you were to, let's say, jump from the building, you're going to start... The force of gravity is going to be acting on you, and you're going to start accelerating downwards. It doesn't seem that the Earth is accelerating up to you. Wouldn't we notice that, every time someone were to jump off a building, that the Earth starting accelerating in a major way. And the way to think about that is that the force is the SAME. And we'll talk about that in other videos. Force is equal to mass times acceleration. So when we are talking about 686 Newtons, in terms of the force of Earth, the gravitational attraction between myself and Earth. And this is going to be 68 kilograms, then this provides a pretty good acceleration on me. So in this case, if you solve for A... Solve for A over here. You're going to get 9.8 meters per second squared. Now, if you do the same thing for Earth... I already told you that we are pulling on each other with the same force, 686 Newtons. Now if you want to find out how much is the Earth accelerating... That force you're going to get a... I'm not even going to put it here. You're going to put a huge number in here. Huge number times the acceleration of Earth towards me. And since this is such a huge number, very very very large number, this is going to be an immeasurably small number, super small number. And frankly it's probably averaged out by the acceleration of Earth, or the force of gravity and all the other people and things on the surface of the planet. So it all averages out in the end. But even if it didn't, it would be negligible, you wouldn't even notice the acceleration of Earth towards me. But you would notice the acceleration of me towards Earth because of our vastly different masses. Even though we have the same force of attraction.