Center of Gravity?

What a great question. I haven't seen an interesting one like that in a while.

Start with a statement of center of mass

x_com=((m_w) x+(m_g)_(h/2))/(m_w + m_g)

Where

x_com=center of mass

m_w=mass of water

m_g=mass of glass

h=height of glass

x=height of water

Since density=mass/Volume

then

mass=density*volume

Since we have a cylinder

Volume=Base area*height

then

mass=density*(base area)*(height of water)

or

m=rho A x

Substitute this into the center of mass equation.

To find a minimum, we take a derivative of our function and set it to zero.

Take a derivative of the center of mass equation using the quotient rule and set the derivative equal to zero.

You then have a messy quadratic equation. Clean it up and then use the quadratic formula.

There are two solutions from the quadratic formula, but only one is positive.

The only simplifications I can see from there is to factor out the 2 and the m_g/(rho A) terms, except if the mass of the glass is large or comparable to the mass of 1/2 a glass of water.

If the mass of the glass is large or comparable to the mass of half a glass of water, then the height for the lowest center of mass comes out the about 1/4 of a glass.

We can get this result by taking a binomial expansion of the root part of the result from the quadratic formula.

I hope this helps.

Find the equation for the center of mass height (y) vs water height (x). This equation will have a minimum value where the slope (derivative) equals zero. Find the zeros of the derivative equation, and choose the one that makes sense. Alternatively, graph the equation and approximate the answer by where the low point in the curve occurs.

Qualitatively, an empty glass with have a COG in the lower half of the glass. Adding water will pull the COG still lower, until the surface of the water is at the same height as the new lowest possible COG. After that, adding more water will raise the COG.

Brian is correct. Unless the glass is constructed strangely, like an upside down pyramid, the glass in the bottom of the glass tilts the balance so that empty it has the lower center of gravity. Glass is more dense than water.

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Well, if you include the glass, then being empty would make the center of mass near the middle bottom of the cup. Adding water to the cup would only raise the center of mass vertically.

Center Of Gravity Philippines