Physics 281: Homework #4: due May 2
Note: Do only one of the following two mechanics
of materials problems.
- If rock has an ultimate stress comparable to that of glass (i.e.,
quartz), how high can a mountain get to be on earth? How does this scale
with planetary radius? At what size is the maximum mountain height
comparable to radius? The answer to the last question indicates at what
size asteroids must begin to appear spherical. It also matters somewhat
whether you consider a cubic or conical shape, so pick the mountain-shaped
one.
- How strong of a wind can a firmly-rooted tree withstand without its
trunk snapping. Note that this can be a failure mode, but the
roots often break and come up instead.
- Fun with Scale Heights: This problem is composed of a number of
mini-problems to increase familiarity with atmospheric physics.
- If we took all the air in the atmosphere, and compressed it to the
density of water, how deep would the layer be? Start with scale height.
This becomes the depth of water at which pressure increases by one
atmosphere.
- The speed of sound goes like the square root of temperature. What is
the effective scale height for speed of sound in the atmosphere?
Commercial airplanes travel at about Mach 0.8 (at altitutde). What speed
does this correspond to (in m/s, km/h and m.p.h.), assuming no wind?
- Create a satellite model for yourself, concentrating on area and mass.
Don't forget that satellites are intentionally lightweight, and have large
solar panels, typically. Put the satellite at a 200 km height and
figure out the reduction in atmospheric density. Despite this being a
staggeringly large reduction, the drag on the satellite is significant. Using the momentum
imparted to the satellite and a relationship between velocity and orbital
radius, calculate how much altitude the satellite loses in a day, spiraling
in on a basically always-circular orbit.
- Invent a problem of your own in the spirit of this week's class topics.
You do not need to provide a complete solution, though you might outline
how it may go. The art is to strike a balance between trivial and hopelessly
complex.