The figure shows two large rocks in outer space. Each rock has a mass of 480 kilograms. What is the magnitude of the gravitational force between the two rocks? Use a value of 6.67 times 10 to the minus 11 meters cubed over kilogram-second squared for the universal gravitational constant. Give your answer in scientific notation to two decimal places. The grid lines on the grid are spaced one meter apart.
On the diagram, we can see our two rocks. And we’re told they’re in outer space, which means we can assume there is nothing else nearby that has mass. And so the only gravitational force that we need to consider is that due to the two rocks acting on each other. We’re also told that each rock has a mass of 480 kilograms. So, we can label these two masses 𝑚 one and 𝑚 two, and they are both 480 kilograms.
We’re asked to find the magnitude of the gravitational force. So, we need to recall that the gravitational force 𝐹 is equal to the universal gravitational constant 𝐺 times the mass of object one 𝑚 one times the mass of object two 𝑚 two divided by the distance between their centers of mass 𝑑 squared. We’re given 𝐺, which is the universal gravitational constant of 6.67 times 10 to the minus 11 meters cubed over kilogram-second squared. And we know that 𝑚 one and 𝑚 two are both 480 kilograms. So, we just need to work out the distance 𝑑 between their centers of mass.
Looking at the diagram, we can see that the distance between the centers of mass is one square to the right and three squares upwards. So, we need to find the length of this side 𝑑, which we can do using Pythagoras’s theorem. Given a right-angled triangle with sides of one meter and three meters, we know from Pythagoras’s theorem that 𝑑 squared is equal to one squared plus three squared, or 𝑑 is equal to the square root of one squared plus three squared. Now, one squared is one and three squared is equal to nine. So, 𝑑 is equal to the square root of 10. And we were told that each one of these grid squares was one meter, so this distance will be the square root of 10 meters. Now, keep in mind that we’re going to square 𝑑. So it’s easiest to keep it in this form as a square root rather than evaluating it as a decimal.
So, substituting numbers into this equation, we have that the gravitational force 𝐹 is equal to 6.67 times 10 to the minus 11, which is the universal gravitational constant 𝐺, times 480, which is mass one, times another 480, which is mass two, times the square root of 10, which is the distance 𝑑, squared. And the square root of 10 squared is, of course, 10. And evaluating these numbers, we get 0.000001537.
That’s not an easy number to read, which is why we’re asked to give this in scientific notation, which means as a number between one and 10 times 10 to some power. We do that by moving our decimal point one, two, three, four, five, six places, which gives us 1.537 times 10 to the minus six. And we’re asked for two decimal places, so that will round up to 1.54 times 10 to the minus six. Then, we need to make sure we’ve used SI units everywhere. So, we have our gravitational constant in meters cubed over kilogram-second squared. The two masses are both in kilograms, and our distance was in meters, which means the units of force will be newtons.
So, the magnitude of the gravitational force between the two rocks is 1.54 times 10 to the minus six newtons.