Video Transcript
An astronaut who has a mass of 80 kilograms goes to Mars. On Mars, the astronaut applies a force of 296 newtons to the ground beneath her due to her weight. What is the acceleration due to gravity at the surface of Mars?
Okay, so in this question, we’ve got an astronaut who’s standing on the surface of Mars. We’re told that this astronaut has a mass of 80 kilograms, which we’ve labeled as 𝑚. We’re also told that she applies a force to the ground beneath her due to her weight. And that force is going to act vertically downward toward the center of mass of Mars. We’ve labeled this weight force as capital 𝑊. And we know that it’s equal to 296 newtons in magnitude. We’re asked to find the acceleration due to gravity at the surface of Mars. And in order to do this, we can recall that there’s a formula which relates this acceleration due to gravity, the mass of an object, and the weight of the object. Specifically, the weight force 𝑊 is equal to the object’s mass 𝑚 multiplied by the acceleration due to gravity 𝑔.
Now, since weight is a force, its SI unit is the newton. Then, on the right-hand side of the equation, the SI unit for mass is the kilogram. Recalling that the units on the left-hand side of an equation must be the same as the units on the right-hand side of that equation, we can see that the quantity 𝑔 must have units of newtons per kilogram. Then, the units on the right-hand side are equal to kilograms multiplied by newtons per kilogram. Since the kilograms and per kilogram cancel each other out, we’re left with units of newtons, which is equal to the units on the left-hand side of the equation.
Now, we said that this quantity 𝑔 is the acceleration due to gravity, but we might also see it referred to as the gravitational field strength. In fact, when we write the quantity 𝑔 with units of newtons per kilogram, we would then typically refer to it as the gravitational field strength. We can recall though that a newton is equivalent to a kilogram-meter per second squared. So then the units of newtons per kilogram can be written as kilogram-meters per second squared divided by kilograms. On the right, the kilograms in the numerator and denominator cancel out. So we have then that units of newtons per kilogram are equal to meters per second squared, where we can recall that meters per second squared is the SI unit for acceleration.
So the quantity 𝑔 can be expressed either in units of newtons per kilogram or equivalently in units of meters per second squared. When we write 𝑔 with these acceleration units of meters per second squared, we then tend to refer to it as the acceleration due to gravity. Since the acceleration due to gravity is what the question is asking us for, then we’ll want to give our answer for 𝑔 with units of meters per second squared. We know that a weight in newtons and a mass in kilograms will mean a value for 𝑔 in units of newtons per kilogram, which is equal to these units of meters per second squared that we want. Since the value we’re given for 𝑊 has units of newtons and our value for 𝑚 has units of kilograms, this means we won’t need to do any unit conversions. And we’ll calculate a value for 𝑔 that has units of meters per second squared.
Before we can use this equation to find the value of 𝑔 though, we’ll need to rearrange it to make 𝑔 the subject. Let’s now clear some space on the screen to do this. To make 𝑔 the subject, we take this equation 𝑊 is equal to 𝑚 multiplied by 𝑔 and we divide both sides of it by the mass 𝑚. On the right-hand side, we’ve got an 𝑚 in the numerator and an 𝑚 in the denominator. So these two 𝑚’s cancel each other out. Then, writing the equation the other way around, we have that the acceleration due to gravity 𝑔 is equal to the weight 𝑊 divided by the mass 𝑚, where we know that if 𝑊 has units of newtons and 𝑚 has units of kilograms, then the value we’ll get for 𝑔 will have units that we can write as meters per second squared.
We’re now ready to substitute our values for the weight 𝑊 and the mass 𝑚 into the right-hand side of this equation. When we do this, we find that 𝑔, the acceleration due to gravity at the surface of Mars, is equal to 296 newtons, which is the weight force 𝑊 exerted by the astronaut on Mars’s surface, divided by 80 kilograms, which is the astronaut’s mass 𝑚.
Now, instead of writing these units of newtons and kilograms individually, we can instead just write the units of meters per second squared that we know that 𝑔 will have. We can then evaluate this expression 296 divided by 80 by typing it into a calculator. When we do this, we get a result of exactly 3.7. And so we found that 𝑔 is equal to 3.7 meters per second squared. This value of 3.7 meters per second squared is our answer for the acceleration due to gravity at the surface of Mars.
