Video Transcript
Europa is the smallest of the four Galilean moons orbiting Jupiter. It has a mass of 4.80 times 10 to the 22 kilograms and a radius of 1,560 kilometers. What is the acceleration due to gravity on the surface of Europa? Give your answer to two decimal places.
In this question, we’re talking about the largest planet in our solar system, Jupiter. In particular, we’re looking at one of its moons. Now, Jupiter actually has many moons, some of which are shown on this diagram, but we’re actually interested in one of the four Galilean moons. The Galilean moons Ganymede, Callisto, Io, and Europa are Jupiter’s four largest moons. And they were discovered by Galileo Galilee in the early 1600s. The smallest of these is Europa. And the question tells us it has a mass of 4.80 times 10 to the 22 kilograms, which we will denote with a capital 𝑀, and a radius of 1,560 kilometers, which we will denote with an 𝑟. The question would like us to work out the acceleration due to gravity on the surface of Europa.
We can work this out by considering the force that acts on a personal object on the surface of Europa. Let’s imagine we have an astronaut that’s landed on the surface that has a mass of 𝑚. Because the astronaut is on the surface of Europa, we know that they are a distance of 𝑟 away from its center of mass. We can work out the gravitational force acting on the astronaut using Newton’s law of gravity. And this law tells us that the magnitude of the force acting on two massive objects is equal to capital 𝐺, the gravitational constant, multiplied by the product of the masses of each of the objects divided by the distance between the object’s center of masses squared.
When we’ve worked out the force on the astronaut, we can then use Newton’s second law, which tells us that the force on an object is equal to the object’s mass multiplied by the acceleration that the object is experiencing. We can rearrange this for the object’s acceleration by dividing both sides of the equation by 𝑚, the object’s mass. And here we can see that the 𝑚’s on the right cancel, leaving us with an expression for the acceleration of the object.
And when we’re considering an object that is being acted on by gravity, we can substitute our expression for gravitational force into this equation, which gives us 𝑎 is equal to 𝐺 multiplied by big 𝑀 multiplied by little 𝑚 divided by 𝑟 squared all divided by little 𝑚. We can now see that the little 𝑚 in the numerator of this fraction and the denominator of this fraction will cancel, giving us an acceleration due to gravity of 𝐺 multiplied by capital 𝑀 divided by 𝑟 squared.
In our case, the acceleration due to gravity is equal to 𝐺 multiplied by the mass of Europa divided by the radius of Europa squared. And because we know 𝑀 and 𝑟 and 𝐺 has a defined value of 6.67 times 10 to the power of negative 11 meters cubed per kilogram second squared, all we have to do is substitute these values into our equation for acceleration, and we’ll get our answer. Substituting these in, we get 𝑎 is equal to 6.67 times 10 to the power of negative 11 meters cubed per kilogram second squared multiplied by 4.80 times 10 to the 22 kilograms divided by 1,560 kilometers squared.
Before we get any further, we should look at our units. In the numerator, we have meters cubed per kilogram second squared multiplied by kilograms. And in the denominator, we have kilometers squared. In the numerator, we’ve used meters to express length, but in the denominator, we’ve got kilometers. To be able to carry out our calculation, these must all be in the same units. So we’ll convert our units of kilometers to meters. We can recall that one kilometer is equal to 1000 meters. Or we could write this as one kilometer is equal to one times 10 to the power of three meters. So we can rewrite our value for the radius of Europa in meters, which is equal to 1,560 times 10 to the power of three meters.
Putting this back into our equation and checking our units again, we can see that these two units of kilograms will cancel out, leaving us with meters cubed per second squared divided by meters squared. We can then bring the meters squared into the denominator of the top fraction, and we can notice that meters cubed is equal to meters multiplied by meters multiplied by meters. And meters squared is equal to meters multiplied by meters. So we can cancel two of these terms from the numerator and denominator of the fraction. This just leaves us with units of meters per second squared. And these are exactly what we’d expect for an acceleration.
So we can continue with our calculation. Evaluating our expression for acceleration, we get 𝑎 is equal to 1.315 and so on meters per second squared. However, the question would like this to two decimal places. So we can just write 𝑎 is equal to 1.32 meters per second squared. And this is our answer. The acceleration due to gravity on the surface of Europa is equal to 1.32 meters per second squared.
