Which of the points 𝐴, 𝐵, 𝐶, and 𝐷 most correctly shows the position of the center of mass of the spring in the diagram. The coils of the spring all have the same uniform density as each other.
Okay, we see this diagram and the four points 𝐴, 𝐵, 𝐶, and 𝐷 marked out on it. The diagram is of a spring, the coils of which all have the same uniform density. We want to figure out which of these four points most correctly shows the position of the center of mass of the spring. To begin figuring this out, let’s recall the definition of the center of mass of an object. The center of mass of an object is that point on it from which the object’s weight acts. We can recall that because weight is a force. It can be represented using a vector.
Say that for this object drawn here, we sketch in that weight force. And say that that force looks like this. Our definition of center of mass tells us that the center of mass of this object we’ve drawn is right there. It’s the point from which the object’s weight acts. Another way we could think of the center of mass of an object is that it’s the average position of matter in the object. So then, we want to find that average position of matter in this spring. Specifically, we want to find which of our four points 𝐴, 𝐵, 𝐶, or 𝐷 is closest to it.
As we do this, one fact that’s helpful is that the coils of this spring all have the same density. So that means that the total mass of this coil, for example, is the same as the total mass of this coil, as this coil, and any other coil in the spring. So to get a sense for the average position of mass in the spring, the point from which the spring’s weight acts, we can count the number of coils. Starting at the top of the spring, we count one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13 coils total.
Now, we saw earlier that each one of these coils has the same mass. So that means the average position of matter in this object, the spring, the point from which the spring’s weight acts, is midway between the sixth coil and the seventh coil. That’s because the total number of coils in the spring, 13, divided by two is six and a half. So if we were to count six and one-half coils from the top or the bottom of the spring, we would find that spot. Starting from the top, we count one, two, three, four, five, six. And here is six and a half.
So on a vertical line through the center of the spring, like the line we see here, the spring’s center of mass will be positioned this elevation above the base of the spring. And we can draw a dashed horizontal line at this elevation to show the line along which that center of mass would occur. Notice that this line is not the same distance from the top of the spring as it is from the bottom. Those distances are different. And the reason is that the density of coils in the spring changes. The coils are more densely packed towards the top of the spring. And they’re less dense towards the bottom. Our line along which the center of mass occurs accounts for this by being closer to the top, the more densely packed area, than towards the bottom.
Looking at our dashed line, we can see that of the four points 𝐴, 𝐵, 𝐶, and 𝐷, our line is closest to point 𝐵. It’s not on this point, but it’s nearer to that point than to any other. And recalling that our question asks for the point that most correctly shows the position of the center of mass, we see that it’s all right that none of them are exactly on it. Our answer, then, is that point 𝐵 most correctly shows that center of mass position of the spring.