A piece of iron is placed 23 centimeters away from a piece of nickel that has a mass of 46 kilograms. Given that the force of gravity between them is 2.9 times 10 to the power of negative eight newtons, determine the mass of the piece of iron. Take the universal gravitational constant 𝐺 is equal to 6.67 times 10 to the power of negative 11 newton meter squared per kilogram squared.
If we take a look at this question, we can see that it talks about the gravitational force between two bodies. It also mentions the universal gravitational constant 𝐺. So therefore, what we can say that is that we’re gonna use Newton’s law of universal gravitation to help us solve the problem. And what we have is that 𝐹 sub 𝐺 is equal to capital 𝐺 multiplied by 𝑚 sub one multiplied by 𝑚 sub two over 𝑟 squared. And this is where 𝐹 sub 𝐺 is the gravitational force. Then we’ve got big 𝐺 or capital 𝐺 is the universal gravitational constant. And what this is here for is to actually make sure that our units line up when we get our final result if we were looking to find the gravitational force.
Then we have our masses. And these are masses of the bodies. And then we have 𝑟, which is our separation or distance between them. Now with this type of problem, what we always do first is have a look at the information we’ve been given. Well, we know the gravitational force 𝐹 sub 𝐺 is 2.9 times 10 to the power of negative eight newtons. Then we know that the mass of the piece of nickel is 46 kilograms. Then we have the mass of the piece of iron, which we’ll call 𝑚 sub 𝐼. Well, this is what we’re trying to find out. So we’ll put a question mark. Then we have the universal gravitational constant, big 𝐺, which is 6.67 multiplied by 10 to the power of negative 11 newton meters squared per kilogram squared. And then finally, we have 𝑟, which is our separation or distance between the two pieces of metal. That is 23 centimeters.
However, on inspection, if we look at the other units that we’re using, we can see, in fact, that we want distances to be in meters. So we’re gonna convert 23 centimeters into meters, which is gonna give us 0.23 meters. Okay, great, so we now have all the information we need to substitute into our formula to help us find the mass of the piece of iron. So before we actually substitute our values in, what we’re gonna do is actually rearrange our formula to make the mass of iron the subject. So we’ve got 𝐹 sub 𝐺 equals 𝐺 multiplied by 𝑚 sub 𝐼 multiplied by 𝑚 sub 𝑁 over 𝑟 squared.
So first of all, if we multiply through by 𝑟 squared, we’re gonna get 𝐹 sub 𝐺 𝑟 squared is equal to 𝐺 multiplied by 𝑚 sub 𝐼 𝑚 sub 𝑁. So then what we do is divide through by 𝐺 𝑚 sub 𝑁. So then what I’ve done here is actually swapped it over so we have the 𝑚 sub 𝐼 on the left-hand side. So we’ve got 𝑚 sub 𝐼, so the mass of the iron, is equal to 𝐹 sub 𝐺 𝑟 squared over 𝐺 multiplied by 𝑚 sub 𝑁. Now all we’re left to do is substitute in our values and calculate the mass of the iron.
So when we substitute in our values, we get the mass of the iron is equal to 2.9 times 10 to the power of negative eight multiplied by 0.23 squared divided by 6.67 times 10 to the power of negative 11 multiplied by 46. So then if we calculate this, we get 0.5. So therefore, we can say that the mass of the iron is 0.5 kilograms.