WEBVTT
00:00:00.200 --> 00:00:03.680
In this video, we’re going to learn about Newton’s law of universal gravitation.
00:00:04.000 --> 00:00:09.000
We’ll see what this law says, how it was developed, and how we can apply it practically.
00:00:09.720 --> 00:00:14.320
To start out imagine that you lived in the time of Isaac Newton in the 1600s.
00:00:15.360 --> 00:00:18.920
During that time, scientists were facing a difficult challenge.
00:00:19.440 --> 00:00:30.360
On the one hand, thanks to astronomical observations, they had clear evidence that planets, celestial bodies they called them, moved around one another in regular orbits.
00:00:31.200 --> 00:00:37.040
Based on this, it seemed clear that there was some sort of attraction between these masses as they moved.
00:00:37.440 --> 00:00:49.400
But on the other hand, on planet Earth with regular everyday size masses, there was no sense that those masses would attract one another even if they were held very close.
00:00:50.520 --> 00:01:06.920
For the time being then, it seemed as though the physical laws governing massive bodies, celestial bodies the size of planets, might have a different form or just be fundamentally different than the laws governing everyday size more tangible objects.
00:01:08.160 --> 00:01:15.320
To see how these two worlds were brought together, it will be helpful to learn about Newton’s law of universal gravitation.
00:01:16.120 --> 00:01:23.840
As the story goes, one day Isaac Newton was resting under an apple tree pondering all these mysteries.
00:01:24.360 --> 00:01:30.000
Then, it said, an apple falls from the tree, hits him on the head, and he has a grand idea.
00:01:30.600 --> 00:01:33.480
There’s a good chance that the story is not accurate.
00:01:34.080 --> 00:01:44.160
But nonetheless, it shows how Newton’s law connects everyday objects we’re familiar with with much larger objects on the scale of the size of our planet or other planets.
00:01:45.240 --> 00:01:50.600
One reason this law is so meaningful is because it truly is universal.
00:01:51.120 --> 00:01:56.680
It applies to any masses no matter how big or small and no matter where they’re located in the universe.
00:01:57.760 --> 00:02:05.160
Considering the scope of this development, it’s remarkable that in one simple statement it’s possible to sum up the gravitational force of attraction between any two masses.
00:02:05.680 --> 00:02:08.480
But that’s just what Newton along with help from his contemporaries did.
00:02:08.880 --> 00:02:18.800
This universal law says that the gravitational force between two objects, mass one and mass two, is equal to their product divided by the square of the distance between them.
00:02:20.000 --> 00:02:22.840
This is the backbone of Newton’s law from a physical perspective.
00:02:23.320 --> 00:02:27.240
And this value is then multiplied by a constant value called big 𝐺.
00:02:28.120 --> 00:02:33.840
This value, called the universal gravitational constant, was developed in order to make the units in the overall expression work out.
00:02:34.440 --> 00:02:37.000
It has its own interesting story of development.
00:02:38.240 --> 00:02:42.840
But before telling that story, let’s consider the rest of this universal law of gravitation.
00:02:43.360 --> 00:02:53.640
Sometimes this law, which is one of the more recognizable equations in physics, can become so familiar to us that we lose sight of what makes it special.
00:02:54.720 --> 00:02:57.720
At the time of its development, this law was not at all obvious though.
00:02:58.360 --> 00:03:02.800
For example, consider the denominator where we see an 𝑟 squared term.
00:03:03.240 --> 00:03:07.120
This means that the universal law of gravitation is an inverse square law.
00:03:07.720 --> 00:03:14.520
Though we see these laws across physics, it’s still a striking result.
00:03:15.080 --> 00:03:16.920
Why one over 𝑟 squared?
00:03:17.400 --> 00:03:22.880
Why not one over 𝑟 cubed or one over 𝑟 to the 1.99?
00:03:23.400 --> 00:03:26.880
Or even why should the force of gravity decrease with increasing distance?
00:03:27.920 --> 00:03:37.360
All of the alternatives we can think of for a one over 𝑟 squared relationship remind us of just how special this law is and how it helps establish the structure of our universe.
00:03:38.160 --> 00:03:41.920
Now, let’s move to considering the mass values 𝑚 one and 𝑚 two.
00:03:42.920 --> 00:03:51.800
This law tells us that if we have two objects of any shape, as long as they have mass, there’s a gravitational force of attraction between them.
00:03:52.200 --> 00:03:56.160
Our masses could be spheres or blocks or crocodiles or atoms.
00:03:57.080 --> 00:04:02.720
Any objects that have mass meet the standard and therefore have a gravitational force between them.
00:04:03.920 --> 00:04:15.080
Regardless of the shapes our two masses have, when we talk about the distance between them, we’re speaking of the distance between their centers of mass wherever those centers are located within the overall mass itself.
00:04:16.120 --> 00:04:18.360
Now, let’s say we try an experiment.
00:04:18.800 --> 00:04:28.160
What if we get two masses and we let their masses both equal exactly one kilogram and we separate these masses by exactly one meter?
00:04:29.200 --> 00:04:50.080
So we can see when we look at this law of gravitation that we’ll have an equation that reads the gravitational force of attraction between these two one-kilogram masses is equal to one kilogram times one kilogram, which is one kilogram squared, all over one meter squared times 𝐺, this gravitational constant.
00:04:50.880 --> 00:04:56.240
In order to show why this gravitational constant is needed, let’s imagine for a second that it’s not there.
00:04:56.640 --> 00:05:02.920
In other words, let’s let it equal one and see what we get as a result of this calculation.
00:05:03.920 --> 00:05:12.520
If big 𝐺 had this value with no units, that would mean that the gravitational force of attraction between these two masses is one kilogram squared meter squared.
00:05:13.360 --> 00:05:23.240
But wait a second, we know that force is measured in units of newtons and that a newton has base units of a kilogram meter per second squared.
00:05:24.000 --> 00:05:29.080
That means the two sides of our equation don’t add up when we let big 𝐺 equal simply one.
00:05:29.920 --> 00:05:33.840
We’re now getting into the whole reason for the existence of the gravitational constant in the first place.
00:05:34.200 --> 00:05:41.400
We’re seeing that if it’s not there — that is, if it’s just equal to one — then our whole expression for the gravitational force of attraction doesn’t make sense.
00:05:42.200 --> 00:05:48.000
Isaac Newton postulated this constant in order to give gravitational forces the right magnitude as well as the right units.
00:05:48.640 --> 00:06:01.800
Gravity as the weakest of the four fundamental forces creates an attractional force between two masses of one kilogram separated by one meter of much much less than one newton.
00:06:02.720 --> 00:06:04.680
So 𝐺 does serve double duty.
00:06:05.360 --> 00:06:10.840
It gives us the right units for our expression as well as a magnitude that agrees with experiment.
00:06:11.800 --> 00:06:20.600
We’ve said that gravity is a weak force compared to the other four fundamental forces of electromagnetism and the strong and weak nuclear forces.
00:06:21.520 --> 00:06:29.520
One way we can see that weakness in action is to take any two household objects we might come across, say a pencil and a water glass.
00:06:30.240 --> 00:06:34.280
If we hold those two objects together, we can’t feel the gravitational force of attraction between them.
00:06:34.560 --> 00:06:35.800
It’s just too weak.
00:06:36.560 --> 00:06:47.080
On the other hand, if we had two magnets, one in each hand, we could definitely feel the force of attraction or repulsion between them as they got close.
00:06:48.040 --> 00:06:52.760
All this is to say that the universal gravitational constant big 𝐺 is a very small value.
00:06:53.440 --> 00:06:54.760
It’s not one.
00:06:54.760 --> 00:06:57.720
In fact, it’s much much less than one.
00:06:58.440 --> 00:07:03.360
It’s so small in fact that it’s very hard to measure big 𝐺.
00:07:03.880 --> 00:07:10.360
One of the most accurate and ingenious measurements of this gravitational constant came under the watch of a gentleman named Henry Cavendish.
00:07:11.280 --> 00:07:15.920
In his experiment, Cavendish suspended a very very thin metal wire from a solid frame.
00:07:16.200 --> 00:07:23.680
From the end of this wire, he hung a small piece of metal that had masses of carefully measured values on either end.
00:07:24.360 --> 00:07:35.280
Once this system had stabilized and wasn’t moving or twisting in anyway, Cavendish brought two relatively large heavy masses close to opposite sides of the smaller suspended masses.
00:07:36.320 --> 00:07:44.400
In response to the gravitational attraction, the smaller suspended masses moved ever so slightly towards the larger ones causing a twist in the wire.
00:07:45.480 --> 00:07:48.480
Cavendish was able to measure that twist with high precision.
00:07:48.880 --> 00:08:02.520
And since he knew all the masses involved as well as the distances separating them, he had values for 𝑚 one, 𝑚 two, 𝑟 as well as the force, 𝐹, the torsional force acting on the wire.
00:08:03.360 --> 00:08:07.960
In other words, he had all the ingredients needed to arrive at a value for 𝐺.
00:08:08.520 --> 00:08:23.120
The value for 𝐺 that Cavendish found is very close to the value that we’ll often use today, which is that 𝐺 is approximately equal to 6.67 times 10 to the negative 11th cubic meters per kilogram second squared.
00:08:24.240 --> 00:08:34.080
Looking at this value, two things perhaps stand out: one, 𝐺 is indeed small much smaller than one and, two, it has a strange set of units attached to it.
00:08:34.880 --> 00:08:41.200
But, remember, the units of 𝐺 are designed to make the rest of the universal law of gravitation have consistent units.
00:08:41.640 --> 00:08:45.360
Measurements for increasingly accurate values of 𝐺 still go on today.
00:08:46.680 --> 00:08:54.040
For our purposes though, we’ll be well served to use this given value for 𝐺, very close to the one that Cavendish found experimentally.
00:08:54.800 --> 00:08:58.920
Let’s get some practice working with Newton’s law of universal gravitation through an example.
00:08:59.720 --> 00:09:04.280
An asteroid has a mass of 4.7 times 10 to the 13th kilograms.
00:09:04.840 --> 00:09:15.720
The asteroid passes near Earth, and at its closest approach, the separation of the centers of mass of the asteroid and Earth is four times the average orbital radius of the Moon.
00:09:16.120 --> 00:09:21.920
What force does the asteroid exert on Earth when at its minimum distance from Earth?
00:09:23.000 --> 00:09:29.720
Use a value of 384400 kilometers for the average orbital radius of the Moon.
00:09:30.680 --> 00:09:36.120
We’ll label this force, we want to solve for, capital 𝐹 and start off by drawing a sketch of the situation.
00:09:37.080 --> 00:09:46.640
In this situation, our asteroid labelled 𝑎 passes by the Earth labelled 𝐸 at a minimum distance of four times the orbital radius of the moon around the Earth.
00:09:47.600 --> 00:09:57.920
Given the mass of the asteroid 𝑚 sub 𝑎 and the orbital radius of the moon around the Earth 𝑂𝑅 sub 𝑚, we want to solve for the gravitational force of attraction between the asteroid and Earth when they’re closest.
00:09:58.960 --> 00:10:12.600
To solve for this force, we recall that the gravitational force of attraction between any two masses, 𝑚 one and 𝑚 two, is equal to their product divided by the square of the distance between their centers of mass all multiplied by the universal gravitational constant capital 𝐺.
00:10:13.520 --> 00:10:21.760
We’ll let that constant 𝐺 be exactly 6.67 times 10 to the negative 11th cubic meters per kilogram second squared.
00:10:22.480 --> 00:10:37.000
When we apply the mathematical relationship for gravitational force to our scenario, we can say that 𝐹, the force we wanna solve for, is equal to 𝐺 times the mass of the Earth times the mass of the asteroid all divided by four times the orbital radius of the Moon quantity squared.
00:10:37.720 --> 00:10:40.800
We know the value in the denominator given to us.
00:10:41.120 --> 00:10:44.200
And we also know the mass of the asteroid, as well as the constant 𝐺.
00:10:44.960 --> 00:10:46.880
All that remains is to solve for the mass of the Earth.
00:10:47.320 --> 00:10:49.600
And we can do that by looking up this value.
00:10:50.400 --> 00:10:57.080
A commonly accepted value for the mass of the Earth is 5.95 times 10 to the 24th kilograms.
00:10:57.600 --> 00:11:02.080
Knowing that, we’re ready to plug in and solve for 𝐹.
00:11:02.360 --> 00:11:10.640
When we do plug these values in, we’re careful to convert the orbital radius of the Moon into units of meters so that it’s consistent with the units in the rest of our expression.
00:11:11.480 --> 00:11:17.520
Speaking of units, let’s take a second to consider what the final units of this calculation will be.
00:11:18.560 --> 00:11:23.640
Looking in the numerator of this overall expression, we see that a factor of kilograms will cancel out.
00:11:24.080 --> 00:11:34.640
Looking then at the units of meters that appear in our expression, we see that the meter squared in the denominator will take away two meter factors in our numerator so that overall we’ll simply have meters to the first.
00:11:35.240 --> 00:11:38.000
The units of second squared will remain in our overall denominator.
00:11:39.040 --> 00:11:48.720
So we expect to get final units of kilograms meters per second squared, which agrees with the units we’d expect for a force, that is, units of newtons.
00:11:49.800 --> 00:11:58.360
When we calculate this result, we find, to two significant figures, that it’s 7.9 times 10 to the ninth newtons.
00:11:58.960 --> 00:12:01.560
That’s the gravitational force of attraction between the asteroid and the Earth.
00:12:02.760 --> 00:12:06.200
Let’s summarize what we’ve learnt so far about Newton’s law of universal gravitation.
00:12:07.440 --> 00:12:14.400
In this video, we’ve seen that Newton’s law of universal gravitation specifies the gravity force between any two masses separated by a distance.
00:12:14.880 --> 00:12:25.720
As an equation, it says that that gravitational force is equal to the product of the masses divided by the square of the distance between their centers of mass all multiplied by this universal gravitational constant called big 𝐺.
00:12:26.840 --> 00:12:35.640
We’ve also seen that gravity is the weakest of the four fundamental forces of gravity electromagnetism and the strong and weak nuclear force.
00:12:36.040 --> 00:12:41.080
And because gravity is such a weak force, we’ve seen that the gravitational constant 𝐺 is difficult to measure.
00:12:42.000 --> 00:12:51.760
However, a working value for 𝐺 has been determined to be 6.67 times 10 to the negative 11th cubic meters per kilograms second squared.