Answer Happy

1. A mass m is attached via a spring with spring
constant k to a wall, with the mass
allowed to move along a horizontal surface. The surface provides a
resistive force proportional to the
velocity Fx = −2mγvx. Using N2, write down a
differential equation describing the displacement of the mass from
its equilibrium position, x(t),
then determine the solution for x(t) in the most general form
possible by assuming
a trial solution x = eλt .
Rewrite the solution from part (a) in amplitude-phase form (you
don’t need to derive this, just state it). Under the assumption
that γ is very small so you can ignore additive terms of
order γ or higher, prove that the total energy in the
system goes down with time.
For the remaining parts, assume the surface is frictionless. A
second mass m is attached to the first mass with another
spring also having a spring constant k. The second mass is not
connected to anything else. Determine a system of coupled
differential equations describing the displacements of the two
masses x1(t) and x2(t). Express these coupled equations
in matrix form.