A cylindrical glass has a mass of 100 grams. It is partially filled with water (density = 1 gram/cubic centimeter.)

The glass has an inside diameter of 8cm and an internal depth of 15cm. When empty the center of mass is 8cm from the top of the glass.

The glass is most stable when its center of gravity is as low as possible. How much water is then in the glass?

(In reply to No Subject by james)

No.  The volumes of glass are not necessarily the same above and below the centre of gravity.  The volumes need to be "weighted" by their distance from the centre of gravity.

If you think about it this way, it may help. You have a glass 100g which may be regarded as a 100g mass at a point 8 cm from the top.  Then the "moment" of the glass about a fixed point x cm from the top is 100 (x-8).  The water volume/mass is given by 16 PI h, where h is the height of the water in the glass.  This acts at a distance 15-h/2-x from the fixed point.

To balance the two: (100) (x-8)=(16 PI h)(15-h/2-x) which we can solve for x in terms of h.

Then we  differentiate and obtain a value for h which maximises (measuring from the top) x.

 

So, Bractals solution is correct.

Posted by goFish on 2005-09-14 12:43:04