Problems 42, 44, 46
Problems 4, 8, 12, 14
Problems 2, 8, 16, 22, 28, 32, 34, 40, 42
Problems 4, 6, 10, 24, 32, 34
Problems 2, 6, 10, 22, 26
Problems 2, 4, 6, 12, 14, 16
Problems 2, 4, 6, 12, 18
Consider the harmonic oscillator
y'' + 23 y = 0.
What is the general solution to this ODE?
What is the
period of oscillation?
What is the frequecy of oscillation?
Pick some initial data. Use maple to plot the solution of the
ODE with this initial data.
In class, we defined the energy
of the solution at time t as:
E(t) = 1/2 m v^2(t) + 1/2 k x^2(t).
Find the initial energy of your solution.
Verify that your
solution has constant energy in time. (Do this by plugging your
solution into the definition of the energy and seeing that it's
constant in time.)
Now add friction to your oscillator. Keep the initial conditions
you chose above but choose three values for the friction beta:
y'' + beta y' + 23 y = 0.
Write down the exact solution for each value of the friction.
each value of the friction, present a plot from maple.
of the values of friction, verify that your solution has decreasing
energy in time. (Again, do this by plugging your solution into
the definition of the energy.)
As time goes to infinity, what does
your energy do? What does this mean physically?
For the case with damping, is it possible for the oscillator to
stop completely in finite time? I.e., is there some time at which
the block stops moving and doesn't start moving again --- it stays
where it is from that time on. If it's possible, find that time.
If it's impossible, prove that it's impossible.