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Newton's Law of Universal Gravitation
M.Ed., George Washington University
Dept. chair at a high school
Matt is currently the department chair at a high school in San Francisco. In his spare time, Matt enjoys spending time outdoors with his wife and two kids.
Newton's Law of Universal Gravitation says that everything with massive particle in the universe attracts all other massive particles. This force is called gravity. The force of gravity = universal gravitational constant (mass of one particle x mass of other particle) / (distance between the two particles)^2.
Newton's universal law of gravitation, remember Newton didn't discover gravity, he just reason that realized the gravity applies to any objects anywhere in space but he also realized that the gravitational force between two objects depended on two things; the first thing he found that, the gravitational force is directly proportional to the mass of each object so too larger objects are going to have a stronger gravitational force than two small objects. He also found that it's inversely proportional to the square of the distance between them. What does that mean? That means that the earth and the sun have a gravitational force on each other, if we have a planet the same size as the earth but its twice as far out, out past Neptune or Pluto, it's going to have a much weaker gravitational force interaction with the sun so with, with those two concepts, he put together a general rule and that was that force is proportional to the mass of the two objects m1 times m2 divided by the square of the distance. Now when we say that force is proportional to that, that means it's not exactly that, it doesn't equal that but actually we have to multiply this unit by a constant in this case it's the universal gravitational constant which we call G and G is going to equal 6.67 times 10 to the negative 11 Newtons meters squared over kilograms squared.
So how do we use Newton's universal gravitation law to solve a problem? Let's take two objects in space that are certain distance from each other and let's determine a gravitational force between 200,000 kilogram objects in space that are each 4,000 meters from each other, from the center to center 4,000 meters apart, okay.
First we notice with G that that is the unit it's so big we've got it in scientific notation so I want convert my whole problem here into scientific notation using exponents. That's going to actually simplify things because instead of multiplying things together, I can just add and subtract those exponents from each other and I'll show you how that's done, so let's first say 100,000 kilograms I want to convert that into scientific units so now I can say that that equals 1, 2, 3, 4, 5 or 1 times 10 to the fifth kilograms so 1 times 10 to the fifth kilograms is a scientific notation for that unit, gets rid of all these zeros. Over here at 4,000 meters again I can go 1, 2, 3 and say that 4,000 meters equals 4 times 10 to the third meters okay? Now I've converted everything to scientific notation it's going to make a little easier to get all these big numbers into this equation okay?
So let's look at the equation force of gravity is G times m1 times m2 divided by the distance squared okay, so we just need to plug those numbers in so my force of G is going to equal again G is 6.67 times 10 to the negative 11 Newton meters squared over kilogram squared okay and we're going to, times m1 squared well I'm going to square it because it's, my mass is the same so it's going to be 1 times 10 to the fifth times 1 times 10 to the fifth so I can just square that and that is over my distance squared. So d squared here is going to be 4 times 10 to the third and let's keep our units in here, we've kilograms there and we have meters there and then I square it okay? So again carrying this down I have 6.67 times 10 to the negative 11 Newton meters squared kilogram squared times and again 1 times 10 to the fifth squared again I can add that so that's going to equal 1 times 10 to the tenth kilograms squared over 4 times 10 to the third, this is going to be 16 times 10 to the sixth and then to keep it in scientific notation I can say 1.6 times 10 to the seventh and my unit there is meters squared okay and separating that right there. Now notice with this constant G I can get rid of some of my units so I have kilograms squared down here I can cancel that with kilograms squared up there. I've got meters squared down here I can cancel with meters squared down there and I have 6.67 times 10 to the negative eleventh times 1 times 10 to the negative tenth so that's going to equal 6.67 times 10 to the negative I'm sorry negative 1 so negative 11 plus 10 is 1 okay? Over and that unit is going to just simply be Newtons over 1.6 times 10 to the seventh and again to move this up here I just make this into negative seventh and I get dividing 6.67 by 1.6 I get 4.16 times 10 to the negative eighth so that is my gravitational force between these two objects.
Now that's a real small number right? That again number is Newtons keeping my units there and that shows the gravitation is really a small force it's just that we're on such a large planet earth that we have a pretty large mass times our mass to make gravitational force seem much stronger on earth but when we're out in space it's relatively weak force.
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