 Hello, and welcome to this screencast. In this screencast, we will calculate the center of mass for an example of a discrete point mass system. We want to find the center of mass of a system that has four point masses. Let's draw a picture of this to see what's happening. Our first mass, M1, weighs 10 pounds and is located at x1 equals zero. All of the weights of the point masses are multiples of 5 pounds, so we'll use two boxes to represent 10 pounds. Next, we have a mass 2, which is equal to 15 pounds, at x2 equals 2, so we'll draw three boxes to represent the 15 pounds. Much further to the right, at x3 equals 7, we have a 5 pound point mass. And then finally, at x equals 8, we have two boxes to represent a 10 pound point mass. We would like to find the center of mass of this system, so if we imagine that these boxes are placed on a teeter-totter like this that will tilt to one side or the other, then finding the center of mass means finding the balancing point of this system. So if the weight were distributed evenly, then the center of mass would be right in the middle of our teeter-totter at x equals 4. However, we have more weight on the left side of this system, so it doesn't seem like it would balance right now. Instead, it looks like the teeter-totter would tilt down on the left side, so even before we make any calculations, we can predict that x-bar, the center of mass where the teeter-totter will balance, is going to have to be less than 4. On the top right of the screen over here, we see the formula for calculating the center of mass. This formula is essentially calculating the weighted average of the point masses, and that weighted average is that balancing point, that center of mass that we're looking for. So first, we will evaluate the sum that we see in the numerator of this formula. For each point mass, m sub i, we multiply its weight by the location x sub i, then we add the results for the four point masses. So our first term is x1 equals 0 times m1 equals 10, then next we have 2 times 15, and then x equals 7 and 5 pounds, and then the last term is our x equals 8 times 10 pounds. And when we simplify this, we get 0 plus 30 plus 35 plus 80, and that numerator then is equal to 145. Next we'll evaluate the sum in the denominator of the formula. This sum is easier because it's just the adding the weights of the four point masses, and so the result is 10 plus 15 plus 5 plus 10, which simplifies to 40. To calculate the center of mass, we divide the 145 by 40, and then we're going to simplify that to get a decimal approximation, and the result is 3.625. Now if we go back to our sketch, we see that x bar equals 3.625 is just a little bit less than 4, and that matches our earlier prediction that we would have to move that balancing point to the left of x equals 4 for our teeter-tartar to balance. And so that gives us our final result, that our center of mass is at x equals 3.625. Thanks for watching.