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.
