Question Video: Finding the Coordinates of the Center of Mass of Three Discrete Masses given Their Coordinates Mathematics

Video Transcript

Three particles are placed on a line. Particle 𝐴 of mass four kilograms is located at the origin; particle 𝐵 of mass six kilograms at nine, six; and particle 𝐶 of mass 10 kilograms at six, four. Determine the coordinates of the center of mass of the three particles.

We will begin by drawing the three particles on a pair of coordinate axes. We are told that particle 𝐴 lies at the origin. Particle 𝐵 lies at the point nine, six. And particle 𝐶 lies at six, four. This question asked us to find the center of mass of the three particles. We recall that the center of mass is in effect the average position of all of the mass in a system. We can calculate the exact position of the center of mass of a system of particles by finding the average position of those particles weighted according to their mass.

Specifically, we can calculate the coordinates of the center of mass of a system consisting of particles in two-dimensional space using these two equations. Firstly, the 𝑥-coordinate of the center of mass, written COM sub 𝑥, can be found by adding together the product of each particle’s mass and 𝑥-coordinate and dividing this quantity by the sum of all the masses of the particles. In the same way, the 𝑦-coordinate of the center of mass, written COM sub 𝑦, can be found by adding together the product of each particle’s mass and 𝑦-coordinate and once again dividing by the sum of all the masses of the particles.

Let’s begin by calculating the 𝑥-coordinate of the center of mass in this question. On the numerator, we need to multiply the mass of particle 𝐴 by the 𝑥-coordinate of particle 𝐴, multiply the mass of particle 𝐵 by the 𝑥-coordinate of particle 𝐵, and multiply the mass of particle 𝐶 by the 𝑥-coordinate of particle 𝐶. We then find the sum of these three values. On the denominator, we have the sum of the masses of the three particles. Particle 𝐴 has a mass of four kilograms and an 𝑥-coordinate of zero as it is located at the origin. Particle 𝐵 has a mass of six kilograms and an 𝑥-coordinate of nine. Finally, particle 𝐶 has a mass of 10 kilograms and an 𝑥-coordinate of six.

The 𝑥-coordinate of the center of mass is therefore equal to four multiplied by zero plus six multiplied by nine plus 10 multiplied by six divided by four plus six plus 10. This simplifies to 114 over 20, which is equal to 5.7. The 𝑥-coordinate of the center of mass is 5.7.

Let’s now consider the 𝑦-coordinate. We calculate this in the same way, except this time on the numerator we multiply the mass of each particle by its corresponding 𝑦-coordinate. This simplifies to 76 over 20, which is equal to 3.8. The 𝑦-coordinate of the center of mass is 3.8.

We can therefore conclude that the center of mass of the three particles has coordinates 5.7, 3.8 as shown.