Michael
J. O’Shea Advanced Mechanics Problems |

Crossing a river with variable current. The optimum
(minimum time) path across a river where the speed of downstream current varies in some complex
way. This is an interesting problem
that can be solved analytically via non-linear differentials equations for a
wide range of current profiles. The
problem can also be mapped onto the well-known mathematical pursuit problem
for simpler cases Elasticity and mechanical advantage. Mechanical
advantage can be obtained by pulling sideways on a fixed rope.
This problem turns out to be mathematically complex but can be solved
exactly for the case where the pull is at the mid-point of the rope. For other cases, numerical solution is
required. An oscillator with harmonic and anharmonic
cross-overs. Related to the previous topic, a
mass at the center of a rope will undergo oscillations if dropped.
This system shows interesting cross-overs between two different types
of harmonic oscillator along with a non-linear regime as a function of
amplitude. In one respect this work is
the dynamic analog of the previous case.
Here the mass is dropped, in the previous case the mass (or force) is
applied and the static properties are examined. |