https://youtu.be/8P-YMHFjHL8
PHYS 1101: Lecture Fourteen, Part Nine
Okay. Newton was the one that worked this out. He arrived at what’s called the Universal Law of Gravitation. Here’s what’s behind the scene, or what’s at the heart of the force due to gravity. All objects that have mass exert gravitational force on each other. The mere fact that they have mass means that there is this interaction between them, this gravitational force.
The situation, though, is that it’s very, very weak. It depends on how much mass each of these two objects have. For typical day-to-day objects, like a television and a laptop, the masses are small. This gravitational force is small because these masses are small. So, you don’t notice this at all. The only time it’s significant is when one of these masses is really big.
Here’s the mathematical equation for the force due to gravity, this Universal Law of Gravitation. The force due to gravity will always have a magnitude that is a constant capital G. It has a value of 6.67 times 10 to the minus 11. Newtons, meters squared, per kilograms squared. It always, universally, has this value.
To that constant, I have to multiply a few factors. I have to look at the product of the two masses that are exerting this gravitational force, and it’s an action/reaction pair. Each of these two masses feels the same size force. They’re equal, opposite direction. I have to include both of these masses, and then I have to divide by the separation of the two. The closer the two masses are, the bigger the gravitational force will be.
Here’s a cartoon to help you visualize what these variables represent. I’ve got two objects, mass 1 and mass 2. Any two objects with any size mass exert a gravitational pull on each other. These two forces are equal and opposite, equal magnitude, opposite direction. m2 feels a pull to the center of m1. m1, as the object, feels a force on it that pulls it to the center of m2. The magnitude of these forces are calculated by the relationship shown right there. Got to multiply G times each mass times r squared. Okay.
Before we apply that to our problem, see if we’ve got enough to calculate the size of fG for the satellite. Let me show you a couple of things. First important point, this gravitational force is only significant if either one or both of these masses is huge, like a planet. That’s why we never include the gravitational force of the trunk of Sarah’s car on the box. There is a gravitational force attraction between the two. There’s also one on the box from Sarah and from all the other mass around her. All of those forces are incredibly small compared to the normal force of the trunk and the weight, the gravitational pull from the Earth.
Okay. So, here’s how we’ve been using this long range gravitational force. Let me point that out. I got a snapshot here of the planet Earth, large mass, and here I am. Looks like Angela is in Africa somewhere, West Africa. Mass of Angela. So, I’m thinking of me as being the object. How large is this force, gravitational force, from the planet Earth? Let’s apply this now general Universal Law of Gravitation to me, and let’s see how it compares to what we’ve been doing for the force due to gravity.
Well, let’s start with… Here’s Angela. What forces are on her? I know there’s the gravitational force from the Earth, and then the surface of the Earth is countering that. So, my normal force is equal and opposite, and it’s balancing me, which is why I’m in equilibrium. I’m just sitting there. We’re going to start this analysis with asking how big is this gravitational force, and let’s start with the Universal Law and see where it leads us.
Newton says that, in general, a gravitational force can be calculated as this constant capital G times the mass of one object, the Earth. Then, times the mass of the object that you care about, that this force is being exerted on. This is the mass of Angela. So, I’m thinking of Angela as the object.
Then, I have to divide by the radius. I’m sorry, the distance between my center and the Earth’s center, and I have to square that.
What would this radius be? Well, I’m, of course, dramatically exaggerated here. I’m actually so small, in reality, on the scale, you wouldn’t see me. This distance between my center and the center of the Earth, effectively, is just the radius of the Earth.
So, this r… I’m going to make it more clear. The separation between these two masses really is just the radius of the Earth. So, what are the numbers for the planet Earth to plug into this? Remember, G was 6.67 times 10 to the minus 11. Newtons, kilogram. Sorry. Newtons, meters squared, per kilogram squared.
I just went to the tables that are in the front of your book. They’re also on your equation sheet pages. To get some other constants here… The mass of the Earth is, of course, huge. 5.98 times 10 to the 24 kilograms. The radius of the Earth is 6.38 times 10 to the 6th meters. So then, we have the numbers for all of these terms, except I’m not going to tell you what my mass is.
Let’s just plug in these numbers and see what we get. If you do that, G, this value, I have to multiply by this mass. I have to divide by the radius of the Earth squared. You end up with 9.8 meters per second squared. That’s the equivalent to everything in yellow.
This is y. This is what, you’ll notice, we’ve been using for the weight. It’s the mass of the object on the surface of the earth, e.g. me, times G, m times G. This is what’s called the surface gravity and has a value of a little g.
We use this for every object, multiplying only by its unique mass, because every object on the surface of the Earth will have the same values for these other terms. If you’re on the surface Earth, this number is the same for every mass. Every mass is going to be much smaller than the Earth itself. So, the radius would basically be the radius of the Earth. That’s why we take this shortcut. We’ve been using… Force due to gravity is always m times little g.