Video: Deducing Changes in the Center of Mass

The diagram shows a drinking glass under three different conditions. The glass is shown empty, and its center of mass is shown at its geometric center. Water is then added to the glass, and the center of mass of the partially filled glass is shown; this is the center of the combined mass of the glass and the water. Finally, an ice cube is placed in the water, and a pencil is used to push the ice cube into the water so it is at rest, with its topmost face just at the height of the water level. Which of the points A, B, C, and D most correctly shows the position of the center of the combined mass of the glass, water, and ice.

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Video Transcript

The diagram shows a drinking glass under three different conditions. The glass is shown empty, and its center of mass is shown at its geometric center. Water is then added to the glass, and the center of mass of the partially filled glass is shown; this is the center of the combined mass of the glass and the water. Finally, an ice cube is placed in the water, and a pencil is used to push the ice cube into the water so it is at rest, with its topmost face just at the height of the water level. Which of the points A, B, C, and D most correctly shows the position of the center of the combined mass of the glass, water, and ice.

All right, let’s look at our diagram which shows this glass under three conditions. First, the glass is completely empty. And we see the location the glass’s center of mass marked out. Then, the glass is partially filled with water. And this water having mass affects the overall center of mass of this system. The centre of mass of the glass plus the water added in is right here. And then, in our last snapshot, we see that an ice cube has been added to this water and then held underwater by our pencil. And we want to know at which of the four points marked out, A, B, C, and D is the new center of mass of this ice-cube-water-and-glass system?

As we get started on this question, let’s clear a bit of space on screen and then recall what the definition of center of mass is. In a constant gravitational field, an object’s center of mass is the point from which the distribution of weight is equal in all directions. So, considering the center of mass of our empty glass, this is the point from which the weight distribution is equal in all directions.

Then, when we add water to the glass, we know that that center of mass will drop downward because now there’s more weight towards the bottom of the glass. And indeed, we see this to be true. The center of mass of this combined glass-and-water system is lower than the center of mass for the glass by itself.

Then, we do something interesting. We add an ice cube to our glass and we hold it underwater. We can see the effect of this from our drawing. The overall level of water in the glass will go up when the ice cube is submerged in it. That makes sense since the ice cube takes up space. If we look at the center of mass of our glass-and-water system before the ice cube was added in, we see the height of that center of mass aligns with point B in our diagram.

And now, with the ice cube added, since water rises in our glass, we know the overall center of mass will also go up. That’s because we now have more mass towards the top side of the glass than we did before the Ice cube was introduced. The only one of our four points with a higher center of mass than point B is point A. And therefore, that’s our answer. This is the new center of mass of our combined glass-and-water-and-ice-cube system. Since we’re raising, or increasing the elevation, of mass in our system, so we also raise the center of mass.

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