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.
