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5Due at the beginning of class, Thursday Oct 21.
1.
(a) Complete the problem of finding the derivative of restoring
force with respect to displacement from equilibrium
for the (symmetric) mooring problem tackled in class on 10/14.
Take it as far as you can analytically.
Suggestion: You might find the problem more manageable (fewer variables)
if you do just one chain. Combining the effects of two chains
is very simple once you have the one-chain system solved.
(b) Estimate (roughly) the mass per unit length of the chain shown in this
image:
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(c) Determine the equilibrium values of all the variables you have introduced
if two such chains of length 400m were used to moor a ship
in water 80m deep, and the anchors are 390m from the ship.
Use Maple's fsolve. The syntax for fsolve is:
fsolve({eqn1,eqn2},{variable1,variable2});
fsolve({eqn1,eqn2},{variable1=guess1,variable2=lower_search_limit..upper_search_limit});
etc.
You will find that fsolve is unsuccessful
if you simply throw all your equations at it at once.
However, you should be able to decouple some of your variables
from the system. For example, the geometrical configuration can
be solved before dealing with tension issues.
(d) Determine the restoring force per meter of displacement
for the case described in (b) and (c) above.
Use Maple's linalg[linsolve].
(d) If the ship was the Exxon Valdez, mass about 200,000 tonnes,
what would be the period of small oscillations about its equilibrium?
(1 tonne = 1000kg)