Physics 303 Test 1


I pledge that I have neither given nor received unauthorized assistance during the completion of this work.


Signature height0pt depth1pt width3in


Questions (5 pts. apiece) Answer in complete, well-written sentences WITHIN the spaces provided.

  1. What is terminal velocity? Explain.

  2. What is the definition of the sine function in terms of complex exponential functions?

  3. What is the difference between the motion of a `plucked', overdamped oscillator and a `plucked', underdamped oscillator?

  4. Consider the figure below which shows the potential energy function for two charged particles. Where are the turning points if the system has total energy $E$ as shown with the dashed line? Explain your choice graphically.

    \includegraphics[height=1.5in]{f1.eps}

Problems. Clearly show all reasoning for full credit. Use a separate sheet to show your work.

1. 25 pts.

A bound particle of mass $m$ moves according to the potential energy

\begin{displaymath} V(x) = V_0 \left ( {a \over x^2 } + {x \over b} \right ) \end{displaymath}

where $V_0$, $a$, and $b$ are positive. Locate any equilibrium points and determine the spring constant $k$ for small oscillations about those equilibrium points. Express your answer in terms of $V_0$, $a$, and $b$.
2. 25 pts.

An athlete can throw a javelin with an initial speed of $v_0 = 20~m/s$ while standing still. The horizontal ($x$) and vertical ($y$) positions of the javelin are

\begin{displaymath} x(t) = v_0 \cos \theta t + x_0 \qquad y(t) = -{g \over 2} t^2 + v_0\sin \theta t + y_0 \end{displaymath}

where $\theta$ is the initial angle of the javelin relative to the horizontal. Suppose the same athlete is now running at a speed of $v_r = 12~m/s$ and throws the javelin with the same strength.

  1. How would you modify the equations above to account for the running motion of the athlete?

  2. At what angle should the athlete throw the javelin to get the greatest distance? Use the results of part a to obtain your answer and assume the the javelin is released from $y=0$ and returns to $y=0$.

3. 30 pts.

Consider the device shown below which could act as a filter to remove pollutants from a liquid. The filter consists of small elliptical holes that have beads of `active' molecules suspended in them that absorb specific chemicals from the liquid. Two such beads are suspended in the filter's holes by using short polymer chains that act as springs.

2.4in

  1. What is the Lagrangian for this system? Only consider motion along the line of the `springs' (longitudinal motion) and ignore the mass of the polymer chains.

  2. What are the equations of motion that describe the system?

  3. What are the eigenvalues or angular frequencies of the system?

\includegraphics[]{f2.eps}