PHYS212 : Homework


Homework Assignment 2 (due 2/2)

Simple harmonic motion, coupled oscillators

  1. A 0.20 kg thin rod of length 1.0 m swings freely from one end. At time $t=0$, it is at an angle of $\theta = 0.050$ rad with respect to vertical, and its angular speed is 0.10 rad/s.

    (a) The general solution to the equation of motion of the system can be written in the forms

    \begin{displaymath}
\theta(t) =
\left\{
\begin{array}{l}
A \cos (\omega t) + B ...
...ega t)\\
Re\{ C e^{i (\omega t + \phi)} \}
\end{array}\right.
\end{displaymath}

    Find the constants $\omega$, $A$, $B$, $C$, and $\phi$ describing the situation described above.

    (b) Sketch a circle in the complex plane representing the trajectory traced by the function $C e^{i (\omega t + \phi)}$. Label $C$ and $\phi$ in the diagram. Also place a dot on the circle corresponding to the time $t=0$.

    \scalebox{0.7}{
\includegraphics{2Springs.eps}
}
  2. Two particles of mass $m$ are connected to each other and to two fixed points by three identical, massless springs of strength $k$ as shown in the diagram. Find the frequencies and amplitude ratios of the normal modes of the system. Consider only longitudinal motion, as we already considered the transverse modes in lecture.

  3. The general solutions to the double pendulum equations of motion can be written as a linear combination of the two normal modes,

    \begin{eqnarray*}
\theta_1 &=& A_+  \cos(\omega_+ t) + B_+  \sin(\omega_+ t) \...
...rt{2} A_-  \cos(\omega_- t)
+ \sqrt{2} B_-  \sin(\omega_- t)
\end{eqnarray*}

    A double pendulum with strings of length 0.50 m is released from rest at $t=0$ with $\theta_{1o} = 0.10$ rad and $\theta_{2o} =
0$ rad.

    (a) What values of $A_+$, $B_+$, $A_-$, and $B_-$ describe this situation?

    (b) Use Maple or a spreadsheet to produce a plot of $\theta_1$, $\theta_2$, and $(\theta_1 + \theta_2)/2$ vs. time over the first few periods of the slow mode. What is the physical meaning of the average?

    (c) What are the angular positions of the masses after four periods of the slower normal mode?

Lab 1: The Double Pendulum


Copyright © 2003-2009, Lewis A. Riley Updated Tue Apr 14 02:04:45 2009