Question Video: Finding the Distance Between Two Objects Given Their Masses and the Gravitational Force Between Them Physics

Video Transcript

Two asteroids 𝐴 and 𝐵 are in deep space. Asteroid 𝐴 has a mass of 5.75 times 10 to the seven kilograms, and asteroid 𝐵 has a mass of 1.39 times 10 to the eight kilograms. If the magnitude of the gravitational force between them is 0.370 newtons, what is the distance between the centers of mass of the two asteroids? Use a value of 6.67 times 10 to the minus 11 meters cubed per kilogram second squared for the universal gravitational constant. Give your answer in scientific notation to two decimal places.

So here we have our two asteroids 𝐴 and 𝐵, and we’re told they’re in deep space, which means there’s nothing with significant mass around and we only have to consider the gravitational force that the two asteroids exert on each other. We need to recall the equation that gives us that force, which is 𝐹, the gravitational force, is equal to the universal gravitational constant 𝐺 times the mass of the first object 𝑚 one times the mass of the second object 𝑚 two divided by the distance between their centers of mass squared.

Now the mass of asteroid 𝐴 is given to us as 5.75 times 10 to the seven kilograms, and the mass of asteroid 𝐵 is given to us as 1.39 times 10 to the eight kilograms. So let’s call that 𝑚 two. We’re also given the value of the gravitational force 𝐹 acting between these objects, which is 0.370 newtons. And then the value we need to find is 𝑑, the distance between the centers of mass of the two asteroids. So we need to rearrange the equation in terms of the quantity we want to find, which is 𝑑. So we’ll start by multiplying both sides by 𝑑 squared and then dividing both sides by 𝐹. So we have 𝑑 squared is equal to 𝐺 times 𝑚 one times 𝑚 two divided by 𝐹, which means 𝑑 is equal to the square root of 𝐺 times 𝑚 one times 𝑚 two divided by 𝐹.

So substituting in the numbers, we have that 𝑑 is equal to the square root of 6.67 times 10 to the minus 11, which is 𝐺, times 5.75 times 10 to the seven, which is 𝑚 one, times 1.39 times 10 to the eight, which is 𝑚 two, divided by 0.370, which is 𝐹. And this all comes to 1,200. Now we’re asked to give this answer in scientific notation, which means expressing it as a number between one and 10 times 10 to some power. So to do that, we need to take our decimal point and move it one, two, three places. And that gives us 1.2 times 10 to the three.

Now we’ve used SI units for everything throughout. We had meters cubed over kilogram second squared for our gravitational constant 𝐺, kilograms for the two masses, and newtons for the force. So the units of our answer will be the SI units of distance, which are meters. So the distance between the centers of mass of the two asteroids is 1.2 times 10 to the three meters.