Video: Relating an Object’s Mass to Its Weight and the Acceleration Due to Gravity at Planetary Surfaces

The force exerted by the Moon’s gravity is six times smaller than the force that exerted by the Earth’s gravity. The weight of an astronaut plus his space suit on the Moon is 250 N. How much does the suited astronaut weigh on Earth? What is the mass of the suited astronaut on the Moon? What is the mass of the suited astronaut on Earth?

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Video Transcript

The force exerted by the Moon’s gravity is six times smaller than the force that is exerted by the Earth’s gravity. The weight of an astronaut plus his space suit on the Moon is 250 newtons. How much does the suited astronaut weigh on Earth? What is the mass of the suited astronaut on the Moon? What is the mass of the suited astronaut on Earth?

Let’s highlight some of this given information. So first, we see that Moon’s gravity is six times weaker than the gravity we experience on Earth, and we also hear that the weight of an astronaut in a space suit is 250 newtons. Then we have three separate questions we want to answer related to this given information. So let’s begin to find the answers to these questions.

As we begin, let’s set up some notational shorthand to use to symbolize different values in this problem. So first, we wanna create a symbol that represents the astronaut’s weight on the Moon; we’ll call that capital 𝑊 sub 𝑚. And as a side note, that’s information we’ve been given; we’re told that that equals 250 newtons.

Now we also wanna create a symbol for the astronaut’s weight on Earth; let’s call that capital 𝑊 sub 𝐸. That’s one of the variables we wanna solve for. Now we also have a term for gravity on Moon and gravity on Earth. Let’s call gravity on Moon lowercase 𝑔 sub 𝑚 and the gravity on Earth lowercase 𝑔 sub 𝑒.

And you may recall from earlier problems that gravity on Earth is taken to be 9.8 meters per second squared. Great, now we have a good start and a good handle for being able to talk about the different quantities we’ll use to solve this problem. So remember, first we wanna solve for 𝑊 sub 𝑒, the weight of the astronaut on Earth.

Now we know in general that weight is a force. One clue to that fact is that if we look at the weight of the astronaut on the Moon, that weight is measured in newtons just like we would measure any other force, so that means we can write an equation which says that the weight of the astronaut on the Moon is equal to the mass of the astronaut on the Moon times the gravity on the Moon, just like that.

And again, we’re told that this product, the mass of the astronaut on the Moon and the gravity on the Moon, is equal to 250 newtons. Now at this point, let’s recall another fact from our problem statement.

The problem said that the gravity on Moon is six times weaker than the gravity on Earth. We can express that as an equation by writing 𝑔 sub 𝑒 equals 𝑔 sub 𝑚 times six. That’s a mathematical way of saying that if we take the gravity on the Moon, however strong that is, we need to multiply that by six to get the gravity on the Earth, which is a known quantity of 9.8 meters per second squared.

Now let’s do this to this equation for gravity; let’s multiply both sides by one over six. And you can see when we do that that the sixes on the right-hand side of the equation cancel out, and we get an expression that says the gravity on the Moon is equal to the gravity on Earth divided by six.

We can now rewrite the mass of the astronaut on the Moon times the gravity on the Moon in terms of the gravity on Earth on our way to figuring out the weight of the astronaut on Earth.

What we find is that the mass of the astronaut on the Moon multiplied by Earth’s gravity divided by six equals 250 newtons. Now as an aside, what is the weight of the astronaut on Earth equal? Well, we know in an equation form the weight of the astronaut on Earth is equal to the mass of the astronaut on Earth times the gravity on Earth.

Now as we write that off to the side and compare that equation with the one we’ve just written, you may see that, in order to get to the weight of the astronaut on Earth, all we would need to do is multiply this lower equation, both sides, by six, so let’s do that.

In doing so, we see the sixes on the left-hand side of the equation cancel out, and we’re left on the right-hand side with the magnitude of six times 250 newtons, which is equal to 1500 newtons.

Now just one more thing to show that 1500 newtons, or 1.50 times 10 to the third newtons, is the weight of the astronaut on Earth. You may say, well wait a second, we had this equation for the weight of the astronaut over here that uses the mass of the astronaut on Earth, 𝑚 sub 𝑒, but in our lower equation on the bottom of the page, we’re using 𝑚 sub 𝑚, the mass of the astronaut on the Moon.

Well here’s something that will help us out enormously to figure this problem out. It turns out that mass is invariant of location. That means that, regardless of the gravitational field in which a mass exists, the mass stays the same.

We can write that relative to our problem by saying that 𝑚 sub 𝑒, the mass of the astronaut on Earth, is equal to 𝑚 sub 𝑚, the mass of the astronaut on the Moon. Having established that relationship, we can now go in and replace 𝑚 sub 𝑚 in our lower equation with 𝑚 sub 𝑒 because they’re the same thing.

We now have an equation that truly symbolizes the weight of the astronaut on Earth. The mass of the astronaut on Earth multiplied by the gravity on Earth is equal to 1.50 times 10 to the third newtons.

Now let’s move on to figure out what exactly is the value of these masses. Alright, let’s take stock of where we are. We just solved for the weight of the astronaut with his space suit on Earth, and we want to now solve for the mass of the astronaut on the Moon and the mass of the astronaut on Earth, and we’ve established that those will actually be the same value, that mass is invariant.

That’s another way of saying that an object’s mass does not change with its location. The object could be on Earth or on the Moon or on Mars or anywhere and its mass will always be the same. Okay, knowing that, let’s write out our force equation for the weight of the astronaut on Earth. The weight of the astronaut on Earth is equal to the mass of the astronaut on Earth multiplied by Earth’s gravity.

Now again, we wanna solve this equation for 𝑚 sub 𝑒, the mass of the astronaut on Earth. To do that, we can multiply both sides of the equation by one over Earth’s gravity. The gravitational field of Earth cancels out on the right-hand side of the equation, and we’re left with an equation which says that the mass of the astronaut and his suit on Earth equals his weight on Earth divided by Earth’s gravity.

Now both the numerator and the denominator here are values we know. We just solved for 𝑊 sub 𝐸, and 𝑔 sub 𝑒, gravity on Earth, is a known value, so let’s plug those values in to this equation.

The mass of the astronaut on Earth equals 1.50 times 10 to the third newtons divided by 9.8 meters per second squared. Performing that division, we find that this number equals 153 kilograms.

Not only does this number equal the mass of the astronaut on Earth, but because of the invariance of an object’s mass, that it doesn’t depend on location, this also equals the mass of the astronaut on the Moon. So by solving for mass once, we’ve solved for it forever; it will always be 153 kilograms.

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