Gravity is an attractive force between two objects that have mass.

Any object that we talk about in this course with the exception of light has mass.

The earth has mass,

I have mass, and you have mass.

There's therefore a gravitational attraction between the earth and me,

the earth and you,

but also between you and I at any given time.

The mathematical description of the force of gravity

needs to take into account the mass of both objects,

and also the distance between them.

In order to get useful information out of any equation,

we also need a universal gravitational constant to

tell us how strong the force will be given the masses and the distances.

Let's call the mass of the larger object capital M,

and the mass of the smaller object little m. The distance between

the two objects will be measured by a lowercase r and the universal gravitational

constant will be denoted as a capital G.

The force of attraction between two objects will

be directly proportional to their masses,

but inversely proportional to the square of the distances separating them.

Direct proportionality means that the force F will be equal to

the universal gravitational constant G times capital M times little m. Finally,

because of the inverse-square relationship,

we divide the whole right hand side of the equation by r to the power of two.

This equation is called Newton's universal law of gravitation,

and calculates the force between two objects no matter what their masses.

In order to use this equation,

we need to consider the units of each term.

G, the universal gravitation constant has a value of 6.67 times 10 to the minus

11 in units of Newton meters squared per kilogram squared, and that's a mouthful.

To make these units cancel out,

you can see that capital M and little m will cancel out the kilograms squared term,

and that the distance squared cancels out the meters squared

term leaving behind Newtons which are a measurement of force.

Notice how tiny the gravitational constant is.

If we ask ourselves how much attractive forces felt between

two objects each weighing one kilogram and separated by one meter,

the answer of course is G times one kilogram,

times one kilogram divided by one meter squared.

So 6.67 times 10 to the minus 11 Newtons or 66.7 picoNewtons.

For comparison 67 picoNewtons is about how hard you have to

pull the two ends of a DNA molecule in order to have them unravel.

But gravity acts on much larger scales and is therefore comparatively weak.

Let's compare 66 picoNewton's to the force of gravity that I feel due to the Earth.

Since Earth weighs 5.97 times 10 to the 24 kilograms and I weigh about 75 kilograms,

in order to calculate the force of gravitational attraction,

we'll replace capital M with Earth's mass and little m with my mass.

We also need to know how far apart the center of the Earth is from the center of me.

Let's take the radius of the Earth's surface to be r and

replace it with a value of 6,378.1 kilometers,

which we have to convert into meters.

So, 6,378,100 meters, which we then square.

Finally, we replaced the universal gravitation constant G

with its value of 6.67 times 10 to the minus 11.

And its units Newton meter squared divided by kilograms squared.

Together, the units of meters cancel each other

out as do the units of kilograms leaving Newtons in the result.

I'll get my calculator out and plug in the math and I get the result of 735 Newtons.

So, I'm being pulled towards the center of the Earth with a force of 735 Newtons.

The unit of force Newtons is sometimes difficult to put into context.

It's related classically with the acceleration of a mass

by Newton's second law F equals ma,

which relates the force on a mass to how quickly the mass accelerates.

Since I feel the force of gravity as 735 Newtons,

I can calculate my acceleration due to gravity by dividing

my mass 75 kilograms which results in acceleration of 9.798 meters per second squared.

You might recognize the coincidence.

The acceleration I feel is very close to

the value of Earth's acceleration due to gravity,

which is often denoted as a little g,

and has an average value of 9.807 meters per second squared.

The reason that these two numbers are different is because

the strength of Earth's gravity varies over a surface.

For example, you weigh about half a percent heavier,

when you're at the Earth's poles than you do when you're along its equator.

In fact, Earth's gravity varies a lot over its surface because

of the different densities of rocks and the different geography of regions.

Earth's gravity diminishes by about one fifth of

one percent from earth's surface to an altitude of five kilometers.

So, your height above or below sea level is also a factor.

But geology can account for another one 100th of a percent difference in gravity.

This map of the globe represents

the difference in Earth's gravity from the average value.

Red indicates stronger gravity and blue indicates weaker gravity.

The data was collected by a pair of satellites called GRACE,

the Gravity Recovery and Climate Experiment.

GRACE uses changes in Earth's gravity to measure changes to

huge masses of ice in our polar regions.

If gravity there decreases,

scientists can determine how much of the glacial ice is melting in those regions and

this data can even tell where vast underground reservoirs of water are filling up.