Project #1: Balancing Act

  1. An open-topped cylindrical cup 20 cm tall and with a 4 cm inner radius is to be filled with a liquid in such a way that it will be most stable. The thickness of the side walls is .1 cm and this part of the cup has a volume mass density of 5 g/cm3. The bottom of the cup is made of a miraculous material of negligible thickness and mass. (You might find it helpful to orient the cup on anxy-axis system with the y-axis serving as an axis of symmetry for the cup.)

a)Comparatively speaking, where do you think the center of mass should be to provide the most stable situation – as high as possible? as low as possible? smack dab in the middle? Or somewhere else?

b)At what height is the center of mass when the cup is empty? When it is full?

c)If we begin filling the cup with water (density 1 g/cm3), how high should we fill it to produce the most stable result?

d)Would there be a difference (other than the environmental risk of a spill) if instead of water we planned to fill the cup with mercury (density 13.5 g/cm3)?

  1. As it turns out, working mundane calculus problems has begun to bore you, so you have joined the equestrian club for entertainment. Their initiation includes a requirement that you be able to trot your filly around the corral carrying a glass of beer on a tray without it turning over. So, now realizing the true value of your calculus education, you turn your attention to the problem of providing the most stable glass of beer by filling it to just the right height. There’s a bit of a catch though, the glass you have to use is a Pilsner glass, shaped as below. How many ounces of beer is required for maximal stability? (Some important details are on the next page.)

  • Here the open interior of the glass is 6” deep and, due to symmetry, the center of mass will lie along the y-axis.
  • The base of the open area is flat and circular and sitting at the level of thex-axis.
  • The interior edge of the glass in the first quadrant is well-approximated by the hyperbola
  • The solid glass base (the part below the axis in the picture) is the frustum of a cone with a lower radius of 1.75” and an upper radius of 1”, and is .5” deep, let’s say.
  • The lateral walls of the glass are .1” thick.
  • The wt. density of glass is approximately .086 lbs. per cubic inch.
  • The mass density of beer is approximately 1010 kg/m3 (you can do the conversion)
  • There are approximately 1.805 cubic inches per fluid ounce. (Better check this figure!)

To complete the project (worth 100 pts.), you need only do one of the above problems.

The maximum possible grade on problem #1 is 100 pts.Due: 4:00 pm October 12

The maximum possible grade on problem #2 is 140 pts.Due: 4:00 pm October 19

The maximum possible grade if you do both is 180 pts.

Feel free to use a calculator liberally, but otherwise your team should do its own work and should show all that work. Be sure to show any integrals that you might use.

If you use any outside sources for information, cite those sources. Do not consult with anyone outside your team – other than me.

There might be information in upcoming classes that could prove useful.