Four particles are positioned at the points zero comma 𝑎, zero comma five, zero comma one, and zero comma three. The center of mass of the four particles is the point 𝐺 is zero comma two. Given that the masses of the four particles are 10𝑚, five 𝑚, four 𝑚, and three 𝑚, respectively, find the value of 𝑎.
To find the value 𝑎 that we’re looking for, we need to combine the information that we have about the locations of the other three particles, the masses of the four particles, and the location of the center of mass. Because we are given the location and mass of each particle, this suggests that we use the center of mass formula, which connects the location and mass of particles to their center of mass. This is the formula for the 𝑦-coordinate of the center of mass indicated by the subscript 𝑦 on the left-hand side.
This coordinate is equal to the sum of the mass of each particle times the 𝑦-coordinate of that particle’s location divided by the total mass of all the particles. Replacing 𝑦 with another coordinate like 𝑥 would give us the formula for the 𝑥-coordinate of the center of mass, although in this case we only care about the 𝑦-coordinate because all of the 𝑥-coordinates involved are zero.
So, we see that the center of mass is effectively the weighted average position of all of the particles, where the weighting is by the mass of the particles. So, the center of mass will tend to be closer to more massive particles and farther from less massive particles. Anyway, what we know from the information that we’re given is the value of all of the masses, the value of all but one of the 𝑦-coordinates, and the 𝑦-coordinate of the center of mass, which means this entire formula actually has only one unknown. So, if we plug in all our values, we should get an equation that we can solve for 𝑎.
For the mass times 𝑦-coordinate portion of the formula, we have 10𝑚 goes with 𝑎, five 𝑚 goes with five, four 𝑚 goes with one, and three 𝑚 goes with three. So, the numerator of our fraction is 10𝑚 times 𝑎 plus five 𝑚 times five plus four 𝑚 times one plus three 𝑚 times three. The denominator, being just the sum of all the masses, is 10𝑚 plus five 𝑚 plus four 𝑚 plus three 𝑚. Finally, this whole expression is equal to the 𝑦-coordinate of the center of mass, which is two.
Let’s now solve this equation for 𝑎. We note that even though we don’t know a value for the unit mass 𝑚, it appears in every term in the numerator and denominator so will divide away and not appear in the final answer. Simplifying the numerator, five times five is 25, four times one is four, and three times three is nine. So we have 25𝑚 plus four 𝑚 plus nine 𝑚, which is 38𝑚. In the denominator, 10 plus five plus four plus three is 22. So, we have 22𝑚. Finally, the right-hand side of this equation is still two.
If we multiply both sides by 22𝑚, on the left-hand side, 22𝑚 divided by 22𝑚 is just one, and we’re left with 10𝑚 times 𝑎 plus 38𝑚. On the right-hand side, two times 22𝑚 is just 44𝑚. Subtracting 38𝑚 from both sides gives us 10𝑚 times 𝑎 equals six 𝑚. Finally, dividing both sides by 10𝑚, on the left-hand side, we get 10𝑚 divided by 10𝑚 is one. So, we just have 𝑎. And on the right-hand side, we have 𝑚 divided by 𝑚 is one. So, we have six over 10 or 0.6. So, 𝑎, the 𝑦-coordinate that we’re looking for, is 0.6.