# Question Video: Finding the Coordinate of the Centre of Mass of Three Discrete Masses given Their Coordinates

Three particles are placed on a line. Particle A of mass 4 kg is located at the origin, particle B of mass 6 kg at (9, 6), and particle C of mass 10 kg at (6, 4). Determine the coordinates of the center of mass of the three particles.

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### 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.

A good place to start with this question is to draw the positions of the three particles on pair of coordinate axes. So here’s particle A at the origin, here’s particle B at coordinate nine, six, and here’s particle C at coordinate six, four. This question askes us to find the center of mass of the three particles. We can recall that the center of mass is effectively 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. The equation on the left tells us that 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 of the masses of the particles. The equation on the right shows us that the 𝑦-coordinate of the center of mass can be calculated in a similar way, only this time we sum the product of each particle’s mass and 𝑦-coordinate. Let’s first apply the equation on the left to find the 𝑥-coordinate of the center of mass of the three particles.

The numerator of this expression tells us that we need to take the mass of each particle multiplied by the 𝑥-coordinate of each particle and then sum these quantities together. In other words, we multiply the mass of particle A, 𝑀 A, by the 𝑥-coordinate of particle A, 𝑥 A, then add the mass of particle B multiplied by the 𝑥-coordinate of particle B and then add the mass of particle C multiplied by the 𝑥-coordinate of particle C. The denominator of this expression is the sum of all of the masses. In this case, that means we’re dividing by the mass of particle A plus the mass of particle B plus the mass of particle C.

Fortunately, all the information we need is available in the question. The mass of particle A is four kilograms, and it’s located at the origin, which means the 𝑥-coordinate is zero. So the mass of A multiplied by the 𝑥-coordinate of A is four times zero. Next, we know that the mass of particle B is six kilograms, and its 𝑥-coordinate is nine. So we add six times nine. Finally, the mass of particle C is 10 kilograms, and its 𝑥-coordinate is six. So, we add 10 times six.

We then divide all this by the mass of A plus the mass of B plus the mass of C, which is four kilograms plus six kilograms plus 10 kilograms. Evaluating the numerator, we have four times zero, which is zero, six times nine, which is 54, and 10 times six, which is 60. 54 plus 60 in the numerator is 114, and four plus six plus 10 in the denominator is 20, which expressed as a decimal is 5.7. So the 𝑥-coordinate of the center of mass of these three particles is 5.7.

Now we just need to find the 𝑦-coordinate of the center of mass. And we can do this using the equation on the right. So first, we multiply each particle’s mass by its 𝑦-coordinate and sum these together. Particle A has a mass of four kilograms, and because it’s located at the origin, we know it has a 𝑦-coordinate of zero. Particle B has a mass of six and a 𝑦-coordinate of six. And particle C has a mass of 10 and a 𝑦-coordinate of four. We then divide all of this by the sum of the masses of all the particles, which we previously calculated to be 20. Now, evaluating the numerator, we have four times zero, which is zero, six times six, which is 36, and 10 times four, which is 40. Summing the values in the numerator gives us 76, and that’s divided by 20, which expressed as decimal is 3.8.

So, if the 𝑥-coordinate is 5.7 and the 𝑦-coordinate is 3.8, then the coordinates of the center of mass are 5.7, 3.8. And this is the final answer to our question.