# Physics 42200 - *Waves and Oscillations*

## Suggestions for Assignment #3

- The first question is asking about a system that acts like an oscillator
and which could potentially resonant with the driving force, if it
is close to the resonant frequency. First, recognize that the
force acting on the log depends on the depth that is submerged, and
this depends on the height of the incident waves. The submerged
depth is not defined in an inertial reference frame, so Newton's
laws do not apply. However, the position of the log in an inertial
frame is just the sum of the height of the wave and the submerged
depth. When you write the differential equation for the submerged
depth, the part that depends on the second derivative of the wave
amplitude can be treated as a driving "force". When determining whether
the log will resonate, calculate Q and γ and use the properties
of the general resonance amplitude curve
to argue whether the amplitude of oscillations will be large or small.
- The second question also involves a non-inertial reference frame:
the force acting on the mass depends on the length of the rubber band,
but the length is defined with respect to the position of the pin on
the wheel, which rotates. Therefore, the pin is not in an inertial
reference frame. To analyze the problem, write the equations of
motion in a frame that is inertial, for example one where all distances
are measured with respect to the axis of the motor.
- This problem might look like it will involve coupled oscillators, but
because the variables chosen to describe the motion are the position
of the center of mass and the angle about the center of mass, they
are actually decoupled. This would not be the case if you had used
the positions of each end of the beam instead.
- Although the driving "force" is just a non-intertial reference frame,
which is normally how you would make a pendulum like this swing, the
problem just requires finding the frequencies of the normal modes
of oscillation. Write the equations of motion for free oscillations
in terms of x
_{1} and x_{2}, assume that the solutions
are proportional to cos(ω t), and solve for ω by calculating
the roots of the characteristic polynomial that comes from the
determinant of the resulting system of equations. Then calculate the
eigenvectors to determine what the two modes of oscillation will look
like.