Video Transcript
Earth has a mass of 5.97 times 10 to the 24 kilograms, and the Moon has a mass of 7.34 times 10 to the 22 kilograms. The average distance between the center of Earth and the center of the Moon is 384,000 kilometers. What is the magnitude of the gravitational force between Earth and the Moon? Use a value of 6.67 times 10 to the negative 11 meters cubed per kilogram second squared for the universal gravitational constant. Give your answer in scientific notation to two decimal places.
In this problem, we are only asked to consider the force between Earth and the Moon. We are told that the mass of Earth, which we’ll call 𝑚 one, is 5.97 times 10 to the 24 kilograms and the mass of the Moon, which we’ll call 𝑚 two, is equal to 7.34 times 10 to the 22 kilograms. We’re also told that the average distance between the center of Earth and the center of the Moon is 384,000 kilometers, so we’ll call this 𝑑.
Note that this is defined as the average distance between the center of Earth and the center of the Moon. This is because the Moon’s orbit around Earth is not perfectly circular. The orbit of the Moon around Earth is an ellipse, so it’s sometimes further away than average. And at other times, it will be closer than average, although this diagram is of course exaggerated and the difference is not significant for the purpose of this problem.
We’re asked to find the magnitude of the gravitational force, so we need to recall the equation 𝐹 is equal to 𝐺 times 𝑚 one times 𝑚 two divided by 𝑑 squared. Here, 𝐹 is the gravitational force. 𝐺 is the universal gravitational constant, which is given to us in the question as 6.67 times 10 to the negative 11 meters cubed per kilogram second squared. 𝑚 one is the mass of Earth, or 5.97 times 10 to the 24 kilograms. 𝑚 two is the mass of the Moon, or 7.34 times 10 to the 22 kilograms. And 𝑑 is the distance between the Earth and the Moon or 384,000 kilometers.
Now, recall that gravitational force always acts from the center of mass. So, if we have two spherical objects with some masses 𝑚 a and 𝑚 b, they will experience some forces 𝐹 a and 𝐹 b along the line connecting the two centers of mass. And these forces will be identical if we consider the two objects as point masses located at their centers. Therefore, the distance we need to put into this equation goes from the center of Earth to the center of the Moon.
One thing that’s useful to do before we put numbers into this equation is to check that we’re using SI base units throughout. The universal gravitational constant 𝐺 is given to us in units of meters cubed per kilogram second squared. So, this is already in the correct units. 𝑚 one and 𝑚 two are both given to us in kilograms, so they’re fine. But the distance between Earth and the Moon is given to us in kilometers. So, recall that one kilometer is equal to 1000 meters, so we’re going to multiply the distance by 1000 and express it in meters.
So, we can now put in our numbers. And we have that the gravitational force is equal to 𝐺, which is 6.67 times 10 to the negative 11 meters cubed per kilogram second squared times 𝑚 one, which is 5.97 times 10 to the 24 kilograms, times 𝑚 two, which is 7.34 times 10 to the 22 kilograms, divided by 𝑑, which is 384 million meters. If we evaluate this, we get that 𝐹 is equal to 1.982 times 10 to the 20. We’re asked to give this in scientific notation, which it already is, and to two decimal places. So, that becomes 1.98 times 10 to the 20.
We now need to work out the units. So, we’ll start with the units of the universal gravitational constant 𝐺, which are meters cubed per kilogram second squared, multiplied by the units of 𝑚 one, which are kilograms, multiplied by the units of 𝑚 two, which takes us to kilograms squared, and divided by the units of distance, which we converted to meters squared. We can cancel out the meter squared in the denominator with two of the meters in the numerator, leaving us with just one meters. And then, the kilograms in the denominator cancel with one of the kilograms in the numerator, leaving us with just one. And we still have second squared. So, that leaves us with meters kilograms per second squared, which is equivalent to newtons.
So, the magnitude of the gravitational force between Earth and the Moon is 1.98 times 10 to the 20 newtons.
