Video Transcript
𝐴𝐵 is a uniform rod of length 27 centimeters and mass two kilograms. Points 𝐶 and 𝐷 trisect the rod such that 𝐶 is closest to 𝐴 and 𝐷 to 𝐵. Given that four masses of eight, four, three, and seven kilograms are placed at 𝐴, 𝐵, 𝐶, and 𝐷, respectively, determine the distance between point 𝐴 and the center of gravity of the system.
Let’s begin with the diagram of the scenario. We have a uniform rod 𝐴𝐵 of length 27 centimeters and mass two kilograms. The rod is trisected, that is, split into three, by points 𝐶 and 𝐷. The distance between 𝐴 and 𝐶 is therefore one-third of 27 centimeters, so nine centimeters. And the distance between 𝐴 and 𝐷 is two-thirds of 27 centimeters, so 18 centimeters. And finally, we have point masses of eight, four, three, and seven kilograms placed at 𝐴, 𝐵, 𝐶, and 𝐷, respectively. Recall that if we have a system of particles, the distance of the center of mass from a particular axis, say the 𝑥-axis, is given by the sum of all of the individual masses, 𝑚 𝑖, times their distance from the axis, 𝑥 𝑖, all divided by the sum of the individual masses.
In this scenario, we have four particles in a system, but we also have a uniform rod. Fortunately, however, we can treat the uniform rod as a particle by finding its center of mass first. Recall that the center of mass of a uniform rod is right at its center. So if the rod is of length 𝑙, then it is 𝑙 over two from either of its ends. To find the center of mass of the whole system then, we can treat the uniform rod as a particle with a point mass of two kilograms right at its center. Since the rod is 27 centimeters long, the distance of this point from the point 𝐴 is 27 over two or 13.5 centimeters.
We now have everything we need to make use of the formula to find the center of mass of the system. We need to go through each of the point masses one by one, multiplying their mass by their distance from the point 𝐴, add them all together, and then divide by their total mass. Starting with the mass at point 𝐴, we have an eight-kilogram mass, and it’s a distance of zero from 𝐴. At point 𝐵, we have a mass of four kilograms, which is 27 centimeters from 𝐴. At 𝐶, we have a mass of three kilograms, which is nine centimeters from 𝐴. At 𝐷, we have a mass of seven kilograms, which is 18 centimeters from 𝐴. And finally, we have the uniform rod which has a mass of two kilograms, and its center of mass is 13.5 centimeters from 𝐴.
Now we need to divide by the total mass, which is eight plus four plus three plus seven plus two kilograms. Adding together the terms on the numerator, we get 288. And on the denominator, we get 24. Simplifying gives us our final answer: The center of mass is 12 centimeters from the point 𝐴.
