Two spherical objects have a combined mass of 150 kg. The gravitational attraction between them is 8.00 x 10⁻⁶ N when their centers are 20 cm apart. What is the mass of each?

Video transcript

Hey, everyone. So this problem is working with gravitational forces. Let's see what it's asking us. Two metallic spheres have a total mass of 210 kg. When the center of these spheres are separated by 18 centimeters, they experience a gravitational pull of nine times 10 to the negative six newtons determine the individual mass of each metallic sphere. In this scenario, our multiple choice answers here are a 23.42 kg and 186.58 kg. B 33.52 kg and 186.58 kg. C 232 sorry 23.42 kg and 191. kg or D 22.42 kg and 182.58 kg. OK. So the first thing we can do here is recall that our gravitational force is given by the equation F equals G M one M two divided by R square. From the problem, we know that both masses equal kg. So we can write that as M one plus M two equals to 10 kg. In other words, M one equals kg minus M two. We look at the other um constants. In this problem G is our gravitational constant. We can recall is 6.67 times 10 to the negative 11 meters cubed per kilogram second squared. And then we are given the force that gravitational force is nine point or sorry, nine times 10, the negative six newtons and our radius or the distance between the two masses is given by 18 centimeters. You're gonna rewrite that as 0. m to keep us in standard units. So from here, we can rearrange our problem to isolate M one and M two are masses. So we'll have write this down here, F R S squared divided by G equals M one multiplied by M two. So when we plug in these known values, we have nine times 10 to the negative six newtons multiplied by 0.18 m squared divided by 6.67 times 10 to the negative 11 max meters cubed per kilogram second squared is equal to and one which we're going to rate in terms of M 2 to kg minus M two multiplied by M two. So from here, we can uh plug in the left hand side of this equation and simplify the right hand side of this equation. And we are left with 4.37 times 10 to the third kilograms squared. It's equal to 210 kg multiplied bye two minus M two squared. We can get this into our quadratic formula notation where we have M two squared minus 210 kg multiplied by M two plus our constant 4.37 times 10 to the third kilogram squared is equal to zero. So we can recall to solve quadratic equations to solve the quadratic equation for M two. We will have negative B or so I can just be be, oh, I'm sorry. No, it is negative because he is negative here. Excuse me guys and gals and folks, OK. Back on track here, negative B plus or minus the square root of B squared minus four AC all divided by two A. OK? So A is going to be one, B is negative 210 and C is 4.37 times 10 to the third. Now when we plug that in to um two to solve for M two, we get 210 plus or minus negative 210 squared minus four multiplied by one, multiplied by 4.37 times 10 to the third. Let's just extend this shit like that all divided by two, multiplied by one or two A. We plug that into our calculator and we get 210 plus or minus 163. divided by two. And so that gives us two, an potential answers for M two uh 186.58, four, 23.4 two. Because the relationship between M one and M two is that the sum of, of those two values equals and two is going to be, you know, choose one of these values and the other one is going to be. So when we look at our potential answer choices that aligns with answer choice A so that is all we have for this one, we'll see you in the next video.