Video: Gravity between Two Spherical Objects

Two objects, object A with a mass of 5 kg and object B with a mass of 100 kg, are near an even larger object with a mass of 10²⁰ kg. Object A and object B are at an equal distance away, 100 km, from the center of mass of the very large object. What is the magnitude of the gravitational force experienced by object A due to the very large object? Give your answer to 3 significant figures.

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Video Transcript

Two objects, object A with a mass of five kilograms and object B with a mass of 100 kilograms, are near an even larger object with a mass of 10 to the 20th kilograms. Object A and object B are at an equal distance away, 100 kilometers, from the center of mass of the very large object. What is the magnitude of the gravitational force experienced by object A due to the very large object? Give your answer to three significant figures.

Okay, looking over at our sketch, we see a five-kilogram object. That must be object A. And as well, we see a 100-kilogram mass. This must be object B. Object A and object B interact gravitationally with this very large spherical object. Now, this object is represented here using a rectangle that’s not very much bigger than A and B. But we can imagine that actually this object is so large it couldn’t fit on the screen at this scale. We have an indication of this thanks to the very large mass of this object.

So, we could almost think of these three objects, A and B and our large spherical one, as though our large spherical object is the Earth, object B, say, is a person, and object A is a book the person is holding on to. That’s roughly the scale we’re talking about as we consider these three objects. Our first question related to this scenario says, what is the magnitude of the gravitational force that object A experiences due to the very large object?

As we get started figuring this out, let’s record some of the information we’ve been given. First, we’ve been told that our large spherical object has a mass, we’ll call it capital 𝑀, of 10 to the 20th kilograms. And we’re also told that the distance between the center of mass of this large spherical object, that is, where all of its mass is effectively concentrated, and our two other objects, object A and object B, is the same. It’s 100 kilometers. We’ll call that distance 𝑟. And as we see, it’s the same for object A and object B in relation to our large spherical object. With these values written down, let’s clear some space and consider again our question.

We want to know the magnitude of the gravitational force experienced by object A, that’s this object here, due to our very large spherical object. So, let’s recall the general equation, called Newton’s law of gravitation, that describes the force of gravity between two objects that have mass. This equation tells us that that force is equal to the product of those masses. We’ll call them capital and lowercase 𝑚, respectively. Multiplied by the universal gravitational constant, capital 𝐺, divided by the distance between our two masses squared.

Now, the gravitational constant, big 𝐺, is approximately equal to 6.67 times 10 to the negative 11th cubic meters per kilogram second squared. So, if we want to know the gravitational force on object A caused by the very large object, we can call this force 𝐹 sub A, then we’ll take this constant and we’ll multiply it by the mass of our large object. And multiply that by the mass of object A, we’ll call it 𝑚 sub A, and divide all this by the distance between the centers of mass of object A and our large spherical object squared.

When we substitute in the values for all these variables, we’re just about ready to calculate this force magnitude, except for one thing. Notice that the distance in our denominator is in units of kilometers. In order to agree with the distance units in the rest of our expression, we’d like to convert this into meters. We can recall that one kilometer is equal to 1000 meters and therefore 100 kilometers is equal to 100,000 meters. With that all figured out, when we calculate this fraction, to three significant figures, we find a result of 3.34 newtons. That’s the magnitude of the gravitational force experienced by object A due to the very large object.

Now, let’s consider the next question in this exercise.

What is the acceleration of object A toward the very large object? Give your answer to three significant figures.

Okay, so, we’ve calculated the gravitational force between these objects. And now, we want to calculate the acceleration object A experiences. We can start doing this by recalling Newton’s second law of motion. This law has it that the net force on an object is equal to that object’s mass times its acceleration. Acceleration, in the case of object A, is just what we want to solve for. And here’s how we can do it. As we saw, Newton’s second law says that the net force on an object is equal to that object’s mass times its acceleration.

When we think about the forces acting on object A, we can see that there are two. One is the gravitational force due to object B and another is the gravitational force due to the large spherical object. But here’s the thing. The force between A and B is much, much, much smaller than the force between A and the large spherical object. This is because the mass of our large object absolutely dwarfs the mass of object B. We can say then that the gravitational force on object A due to the very large object is effectively the only force on object A.

And this means we can say that this equation is equal to the mass of object A multiplied by the acceleration of object A. We’ll call it 𝑎 sub A. And note that we got this expression here from Newton’s second law. But now, as we consider this equation, notice that the mass of object A appears on both sides, and therefore we can cancel it out. If we do that, this is the equation that results. The acceleration of object A is equal to the universal gravitational constant times the mass of our very large object divided by the distance between the centers of mass of object A and our very large object squared.

Now, as we consider this part of our expression here, we can see that it’s actually equal to this expression if we drop out the mass of our object, in this case, object A. With that mass gone, see that we have the gravitational constant, uppercase 𝐺, multiplied by the mass of our large object divided by 𝑟 squared. Which are identical to the factors we see here on the right-hand side of this equation. So then, if we calculate all this, we’ll solve for the acceleration of object A due to the very large object. To three significant figures, it’s 0.667 meters per second squared. That’s the acceleration of object A toward the very large object.

Now, let’s consider the next part of our exercise.

What is the magnitude of the gravitational force experienced by object B due to the very large object? Give your answer to three significant figures.

Okay, whereas before, we were considering the gravitational force between object A and the large spherical object, now we’re considering that force between the large object and object B. Once again, we’ll use Newton’s law of gravitation to solve for this force. But this time, instead of solving for the force experienced by object A, we’ll solve for that experienced by object B. And we’ll say that this object has a mass 𝑚 sub B. Because both objects A and B are the same distance away from the center of mass of our large spherical object, we won’t change 𝑟, the distance between the centers of mass of the two masses in our equation. Which means that when we go to calculate this force, 𝐹 sub B, using an expression like this, all we’ll need to do is substitute the mass of object B in where we used to have the mass of object A.

We see that this mass is 100 kilograms. And so, now, we have an expression which, when we solve it, will give us this force experienced by object B due to the gravitational attraction from the large spherical object. To three significant figures, this force is 66.7 newtons. Note that this is larger than the force magnitude experienced by object A. In fact, it’s 20 times larger because object B has a mass 20 times larger than object A.

Now that we found this out, let’s consider the last part of our question.

What is the acceleration of object B toward the very large object? Give your answer to three significant figures.

To answer this question, once again, we’ll use Newton’s second law of motion. And just like we did for object A, we’ll also assume that the gravitational force between objects A and B is negligibly small compared to that force between object B and the very large object. This means we can say that the mass of object B multiplied by its acceleration is equal to the gravitational force experienced by object B due to the very large object. So then, we can go to our equation for that force, and we can equate it to the mass of object B times the acceleration of object B. We’ll call it 𝑎 sub B.

And then, like we saw earlier, the mass of our object, in this case, object B, is common to both sides. And that means that the acceleration experienced by object B is equal to capital 𝐺 times the mass of our very large object divided by the distance between B and our very large object squared. And this expression here is equal to this expression here if we ignore this mass of our object B. And so, to solve for 𝑎 sub B, we’ll multiply the universal gravitational constant by the mass of our large object divided by the distance between that object and B squared.

To three significant figures, this is 0.667 meters per second squared. Note that this is the same as the acceleration experienced by object A. And this comes back to the fact that our equation for object acceleration does not depend on the mass of the object whose acceleration we’re considering. So, this is how object B and object A, and any other object, will accelerate toward the very large object.

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