Question Video: Finding the Centre of Gravity for a System of Two Masses on the Same Axis Mathematics

Video Transcript

Two particles of weight eight newtons and 18 newtons are separated by a distance of 39 meters. Find the distance between the particle of weight eight newtons and the center of gravity of the system.

The center of gravity of a system is the average position of the components in that system, weighted by their weights. So in general, the center of gravity will tend to be closer to the heavier components and farther from the lighter components. We can calculate the position of the center of gravity with a formula that is almost identical to the formula for the center of mass. In particular, to find the 𝑥-coordinate of the center of gravity, for each component, we multiply its weight by its 𝑥-coordinate and then add all of those together and divide that sum by the total weight of the system.

To find the center of gravity of any other component, we simply replace the 𝑥 with that component, say 𝑦. And to turn this into the center of mass formula, we simply replace each weight with the corresponding mass. Now we aren’t told the coordinates of these two particles, only their weights and the distance between them. But this is okay, since all we’re looking for is a distance, not a set of coordinates. Furthermore, since two points always lie on a straight line, we may as well have that line be horizontal, since, again, we don’t actually care about the particular coordinates, just the distance.

Here, we’ve drawn our two particles at the two ends of a horizontal line. The advantage of having a horizontal line is that it means we only need to calculate the center of gravity in the 𝑥-coordinate. So we only need to do one calculation. But what are the 𝑥-coordinates of these two particles? All we know is that the distance between them is 39 meters. It actually doesn’t matter what the 𝑥-coordinates are. If the 𝑥-coordinate of the eight-newton particle is 𝑎, then the 𝑥-coordinate of the 18-newton particle will be 𝑎 plus 39.

When we use these two 𝑥-coordinates and the two weights to plug into the center of gravity formula, we will find that the center of gravity is located somewhere over here closer to 18 newtons than it is to eight newtons. But the 𝑥-coordinate of this position will be 𝑎 plus some number 𝑑, where 𝑑 is independent of 𝑎. But that means that the distance from the eight-newton particle to the center of gravity, which is what we’re looking for, will be 𝑎 plus 𝑑 minus 𝑎, which is just 𝑑. So since the answer will be 𝑑, regardless of the value that we pick for 𝑎, we can pick whatever value for 𝑎 will make our calculations easiest. We may as well pick 𝑎 equals zero.

If this is the case, then the 𝑥-coordinate of the eight-newton particle is zero, and the 𝑥-coordinate of the 18-newton particle is zero plus 39 or just 39. There are two advantages to this choice. First, since anything times zero is still zero, having an 𝑥-coordinate of zero will eliminate one term from the numerator of our sum. Secondly, the result that we’ll get, 𝑎 plus 𝑑, will instead be zero plus 𝑑 or just 𝑑. So by letting the eight-newton particle be at the position 𝑥 equals zero, we will have not only simplified our center of gravity equation. The result that we get will be exactly the distance that we’re looking for.

All right, let’s plug in some values. In the numerator, we have eight newtons time zero meters plus 18 newtons times 39 meters. And in the denominator, we have eight newtons plus 18 newtons. Eight times zero is zero and 18 times 39 is 702. So the numerator is just 702. And in the denominator, eight plus 18 is 26. 702 divided by 26 is 27. And since the numerator had units of newtons times meters while the denominator had units of just newtons, this answer has units of meters. So 27 is the 𝑥-coordinate of the center of gravity of our system. But as we saw, this 𝑥-coordinate is also exactly the distance that we’re looking for. So the particle with weight eight newtons is 27 meters away from the center of gravity of the system.