# Physics 42200 - Waves and Oscillations

## Suggestions for Assignment #3

1. 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.
2. 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.
3. 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.
4. 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 x1 and x2, 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.