Physics 303 Test 1


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Questions (5 pts. apiece) Answer questions 1-3 in complete, well-written sentences WITHIN the spaces provided. For multiple-choice questions 4-5 circle the correct answer.

  1. Consider the figure below. Where is the equilibrium point? How did you determine it?

    \includegraphics[height=1.75in]{f1.eps}

  2. Why do we use the harmonic oscillator potential when we are approximating the potential energy of a bound system? Explain.

  3. Can we integrate the equation below? Why or why not?


    \begin{displaymath} \frac{dv}{dt} = \sin \theta \end{displaymath}

  4. When a $4.0~kg$ mass is hung vertically on a light spring that obey's Hooke's Law, the spring stretches $2.0~cm$. How much work must an external agent do to stretch the spring $4.0~cm$ from it's equilibrium position?

    A. 1.57 J B. 0.39 J
    C. 0.20 J D. 3.14 J
    E. 0.78 J    

  5. A $100~g$ mass attached to a spring moves on a horizontal frictionless table in simple harmonic motion with amplitude $16~cm$ and period $2~s$. Assuming that the mass is released from rest at $t=0~s$ and $x=-16~cm$, find the displacement as a function of time.

    A. $x=16 \cos (\pi t)$

    B. $x=-16 \cos (\pi t + \pi)$

    C. $x=16 \cos (\pi t + \pi)$

    D. $x=-16 \cos (2\pi t + \pi)$

    E. $x=-16 \cos (\frac{\pi t}{2})$


    \includegraphics{f4.eps}




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

1. 25 pts.

For the damped oscillator the equation of motion is

\begin{displaymath} \ddot y + 2\gamma \dot y + \omega_0^2 y = 0 \end{displaymath}

where $\gamma$ and $\omega_0$ are constants and $y$ is the distance from equilibrium. For $\gamma^2 < \omega_o^2$ the general solution is

\begin{displaymath} y(t) = c_1 e^{(-\gamma + i\Omega^\prime)t} + c_2 e^{-(\gamma + i\Omega^\prime)t} \end{displaymath}

where $\Omega^\prime = \sqrt{\omega_0^2 - \gamma^2}$. Apply the following boundary conditions
 \begin{displaymath} {\rm for}\ t=0 \Longrightarrow \ y = y_0 \ {\rm and} \ \dot{y} = 0 \end{displaymath} (1)

to determine the constants $c_1$ and $c_2$ in terms of $\Omega^\prime$, $\omega_0$, $\gamma$, $y_0$, and any other appropriate parameters.
2. 25 pts.

A boat is slowed by a drag force $F(v)$. Its velocity decreases according to the formula

\begin{displaymath} v = d^3 (t-t_1)^3 \end{displaymath}

where $c$ is a constant and $t_1$ is the time at which it stops. Find the force $F(v)$ as a function of $v$.

3. 25 pts.

A mass $3m$ is suspended from a fixed support by a spring with spring constant $2k$. A second mass $m$ is suspended from the first mass by a spring of spring constant $k$. Use Lagrangian methods to find the equations of motion for this system. Neglect the masses of the spring. Hint: It is easiest to choose the coordinates of the two masses at their equilibrium positions.


\includegraphics{f2.eps}




Equations, Conversions, and Constants


\begin{displaymath} \vec F = m \vec a = \dot {\vec p} = - \nabla V \qquad \vec F... ...ver r^2} \hat r \qquad \vec F_C = {k q_1 q_2 \over r^2} \hat r \end{displaymath}


\begin{displaymath} \vec F_g = - m g \hat y \qquad \vec F_s = - k r \hat r \qquad \vec F_f = -b v \hat v \qquad \vec F_f = - c v^2 \hat v \end{displaymath}


\begin{displaymath} \int {df \over dx } dx = \int df \qquad \ddot y + A \dot y +... ...y + \omega_0^2 y = 0 \Rightarrow y = A \sin(\omega_0 t + \phi) \end{displaymath}


\begin{displaymath} V = - \int_{x_s}^x \vec F(\vec r^{\ \prime}) \cdot d\vec r^{... ...d V_G = -{Gm_1 m_2 \over r} \qquad V_C = - {k q_1 q_2 \over r} \end{displaymath}


\begin{displaymath} K = {1 \over 2} mv^2 \quad L = K - V \qquad {d\ \over dt}\le... ... \dot q} \right ) - {\partial L \over \partial q} = 0 \qquad \end{displaymath}


$\rho$ (water) $1.0\times 10^3 kg/m^3$ $P_{atm}$ $1.05\times 10^5 ~N/m^2$
$k_B$ $1.38\times 10^{-23}~J/K$ Speed of light ($c$) $3.0\times 10^{8}~m/s$
$R$ $8.31J/K-mole$ $g$ $9.8~m/s^2$
$0~K$ $\rm -273^\circ~C$ $1 ~ u$ $1.67\times 10^{-27}~kg$
Gravitation constant ($G$) $6.67 \times 10^{-11}~N-m^2/kg^2$ Earth's radius $6.37\times 10^6~m$
Coulomb constant ($k_e$) $8.99\times 10^{9} {N-m^2 \over C^2}$ Earth's mass $5.97\times 10^{24}~kg$
Elementary charge ($e$) $1.60\times 10^{-19}~C$ Proton/Neutron mass $1.67\times 10^{-27}~kg$
Planck's constant ($h$) $6.626\times 10^{-34}~J-s$ Proton/Neutron mass $932\times 10^{6}~eV/c^2$
Permittivity constant ($\epsilon_0$) $8.85\times 10^{-12} {kg^2\over N-m^2}$ Electron mass $9.11\times 10^{-31}~kg$
Permeability constant ($\mu_0$) $4\pi\times 10^{-7} N/A^2$ Electron mass $0.55\times 10^{6}~MeV/c^2$
$\rm 1 ~MeV$ $10^6 ~ eV$ $\rm 1.0~eV$ $1.6\times 10^{-19}~J$