Two objects, 𝐴 and 𝐵, are in deep space. Object 𝐴 has a mass of 15,000 kilograms and object 𝐵 has a mass of 26,000 kilograms. The distance between the centers of mass of the two objects is 25 meters. What is the acceleration of object 𝐵 toward object 𝐴 due to their gravitational interaction? Use a value of 6.67 times 10 to the minus 11 meters cubed over kilogram second squared for the universal gravitational constant. Give your answer in scientific notation to two decimal places.
So here are our two objects 𝐴 and 𝐵. We’re told that object 𝐴 has a mass of 15,000 kilograms, so we’ll call that 𝑚𝐴, and object 𝐵, we’re told, has a mass of 26,000 kilograms. So we’ll call that 𝑚𝐵. We’re also told that the distance between the centers of mass of the two objects is 25 meters. So let’s call that distance 𝑑. We’re also told that the two objects are in deep space. This means there is nothing around that has any significant mass, and so the only forces we need to consider are those due to the mass of the two objects.
And we know from Newton’s law of gravity that both objects will experience a gravitational force due to the mass of the other. Gravitational force always attracts, which means that this force will act to pull the two objects together. And the gravitational force always acts along the line connecting the objects’ centers of mass. Because of this gravitational force, both objects will experience some acceleration towards the other. We could call the acceleration of object 𝐴 𝑎 sub 𝐴 and the acceleration of object 𝐵 𝑎 sub 𝐵. And 𝑎 sub 𝐵 is the one that we’re trying to find here.
Now we need to recall the relationship between a force and the acceleration due to that force. That is 𝐹 equals 𝑚𝑎, where 𝐹 is the force, 𝑚 is the mass of the object, and 𝑎 is the acceleration that object experiences due to the force. Now, in this case, we’re only asked to find the acceleration due to the gravitational interaction between the objects. So the force here is going to be the force due to gravity. So this force will be equal to the universal gravitational constant 𝐺 times the mass of object one times the mass of object two divided by the distance between their centers of mass squared.
In this case, we’re trying to calculate the acceleration of object 𝐵. So the relevant mass here is that of object 𝐵, which we’ve called 𝑚 subscript 𝐵. And that will be multiplied by the acceleration of object 𝐵, which is 𝑎 subscript 𝐵. And this is equal to the universal gravitational constant times the masses of the two objects, which we’ve called 𝑚 subscript 𝐴 and 𝑚 subscript 𝐵, divided by the distance between their centers of mass, 𝑑 squared.
The first important thing to note here is that the mass of object 𝐵 appears on both sides, which means those cancel out. And we’re left with the acceleration of object 𝐵, which is equal to the universal gravitational constant 𝐺, times the mass of object 𝐴 divided by the distance between them squared.
So putting numbers into this, we have 𝐺 is equal to 6.67 times 10 to the minus 11 multiplied by the mass of object 𝐴, which is 15,000 kilograms, divided by the distance of 25 meters squared. And once we’ve put all these numbers in, we get an answer of 0.0000000016008. And that’s quite difficult to read, which is why we’re asked to give the answer in scientific notation, which means as a number between one and 10 times 10 to some power.
We can do that by moving the decimal point one, two, three, four, five, six, seven, eight, nine spaces, which gives us 1.6008 times 10 to the minus nine. And then we’re asked for two decimal places. So that becomes 1.60 times 10 to the minus nine.
Now for the units we’ve used SI units everywhere. We had meters cubed over kilogram second squared for 𝐺, the mass of object 𝐴 was given in kilograms, and the distance in meters, which means that our answer will be in the SI units of acceleration, which are meters per second squared. So the acceleration of object 𝐵 due to its gravitational interaction with object 𝐴 is 1.60 times 10 to the minus nine meters per second squared.
Now one important thing to notice about this problem is that we didn’t use the mass of object 𝐵 at all because it canceled out in this equation. So that leads us to the important conclusion that the acceleration of an object due to gravity does not depend on its mass. It only depends on the mass of the object whose gravitational field it’s experiencing, in which case that was 𝑚𝐴, and its distance from that object, which means this acceleration would have been the same for any object of any mass placed 25 meters away from the center of object 𝐴.