Physics 132-2 Final Exam


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

  1. Recall the laboratory on Galilean relativity. A projectile was fired from a toy cannon sitting on a cart that was moving horizontally. For the horizontal component of the projectile's motion, what is the relationship between the stationary reference frame ($x$) and the frame of reference moving along with the cart ($x^\prime$)? Clearly explain the components of the relationship.








  2. Consider a container of water with ice added to it and placed on a heater. What is the relationship between the temperature and the added heat after the ice has melted, but before the water begins to boil? What is your evidence?








  3. We measured the heat of vaporization of nitrogen by measuring the rate of the mass loss from a cup of liquid nitrogen hanging from a force sensor as a small resistor heated the liquid. We also measured the mass loss with no current in the resistor before and after we made the measurements of the mass loss with the resistor turned on. Why?








  4. Recall the laboratory on the kinetic theory of ideal gases where you calculated the specific heat of an ideal gas. We claimed this calculation supported the atomic theory. How?








  5. Consider the plot below of the entropy of two Einstein solids $S_A$ (green) and $S_B$ (blue) and their combined entropy $S_{total}$ (red) plotted as a function of the number of energy quanta $q_A$ in solid $A$. On the plot label the equilibrium state of the combined Einstein solids. Explain your choice.



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




  6. Find some equipotential surfaces for the electric dipole charge configuration shown below. Explain your method for finding the equipotentials.



    \includegraphics[height=1.25in]{f2.eps}




  7. Identify a point on the figure below where the magnetic field is zero. Explain how you picked the point.



    \includegraphics[height=1.25in]{f3.eps}

  8. Consider some results from our laboratory on radiocarbon dating. The table shows the number of counts and the uncertainty for different types of radiation and different types of shielding. Which type of radiation is most penetrating? Why?


    Radiation    Air       Plastic     Lead   
    \( \gamma \) $302\pm 18$ $267\pm 16$ $229\pm 15$
    \( \beta \) $117\pm 11$ $43\pm 7$ $17\pm 4$
    \( \alpha \) $19\pm 4$ $12\pm 3$ $13\pm 4$


  9. If you hold two flashlights close together will you see an interference pattern? Explain.








  10. What happens to the width of the central maximum of a single-slit diffraction pattern as the width of the slit is made narrower? Explain.








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

1. 5 pts. How fast must a meter stick be moving if its length is measured to shrink to $0.40~ m$?

2. 7 pts.

The fraternal twins Xena and Thor join a migration from Earth to Planet 10. Planet 10 is a distance $25 ~ly$ away in an inertial reference frame at rest with respect to both planets. The twins of the same age depart at the same time on different spaceships. Xena's ship has a speed $v_X = 0.90c$ while Thor travels at $v_T = 0.80 c$. What is the age difference between the twins when Thor arrives at Planet 10? Which twin is older?

3. 8 pts.

An electron is released from rest on the axis of a uniformly charged ring a distance $x = 0.2~m$ from the center of the ring. If the linear charge density of the ring is $\lambda = 1.5\times 10^{-7}~C/m$ and the radius of the ring is $r=0.25~m$, what is the potential energy of the electron at the moment of release?

4. 10 pts.

A newly-created material has a multiplicity

\begin{displaymath} \Omega = \beta N E e^{\alpha N E} \end{displaymath}

where $N$ is the number of atoms in the solid, $E$ is the total, internal energy in the solid, and $\alpha$ and $\beta$ are constants of proportionality. (a) How does the temperature of the new material depend on the internal energy? (b) What is the molar heat capacity for this solid?

5. 10 pts.

A proton is at rest at the plane vertical boundary of a region containing a uniform vertical magnetic field $B$. A $\rm ^3He$ particle moving horizontally makes a head-on elastic collision with the proton. Immediately after the collision, both particles enter the magnetic field, moving perpendicular to the direction of the field. The radius of the proton's trajectory is $R_p$. Find the radius of the $\rm ^3He$ particle's trajectory in terms of $R_p$. The mass of the $\rm ^3He$ particle is three times that of the proton and its charge is twice that of the proton. The proton charge is $e$.

6. 10 pts.

A room with a volume $V_0 = 100 ~m^3$ is filled with an ideal diatomic gas (air) at a temperature $T_0 = 283~K$ and pressure $P_0 = 1.0\times 10^5~N/m^2$. The air in the room is heated to a new temperature $T_1 = 297~K$ with the pressure remaining at $P_0$ since the room is not airtight. What is the initial internal energy of the air in the room? What is the change in the internal energy of the air in the room? Does this result make sense? Explain. Notice the room is not airtight so air can move freely in and out of it.

Some constants and conversion factors.

$\rho$ (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 $1.67\times 10^{-27}~kg$
Permittivity constant ($\epsilon_0$) $8.85\times 10^{-12} {kg^2\over N-m^2}$ Electron mass $9.11\times 10^{-31}~kg$
$\rm 1 ~MeV$ $10^6 ~ eV$ $\rm 1.0~eV$ $1.6\times 10^{-19}~J$

Physics 132-2 Equations


\begin{displaymath} P = {\vert\vec F\vert \over A} \qquad PV = Nk_B T = nRT \qqu... ...lta T = cm\Delta T = n C_v \Delta T \qquad Q_{f,v} = m L_{f,v} \end{displaymath}


\begin{displaymath} \Delta E_{int} = Q+W \qquad W = \int_a^b \vec F \cdot d\vec ... ..._b} P dV \rightarrow P\Delta V \qquad W = F_{constant}d \qquad \end{displaymath}


\begin{displaymath} \vec F = m \vec a = {d\vec p \over dt} \quad \langle \vec F ... ...= ME_1 \quad \vec p = m \vec v \quad \vec p_0 = \vec p_1 \quad \end{displaymath}


\begin{displaymath} x = \frac{a}{2} t^2 + v_0 t + x_0 \qquad v = at + v_0 \qquad a_c = \frac{v^2}{r} \end{displaymath}


\begin{displaymath} \overline {KE} = \langle E_{kin} \rangle = {1 \over 2}m \o... ...qquad E_{int} = N ~ \langle E_{kin} \rangle = {3\over 2} Nk_BT \end{displaymath}


\begin{displaymath} v_{rms} = \sqrt{\overline {v^2} } \qquad C_{V} = {f\over 2} ... ...\qquad E_f = { k_BT \over 2} \qquad E_{int} = {f\over 2} Nk_BT \end{displaymath}

f $\equiv$ number of degrees of freedom


\begin{displaymath} E_{atom} = (n_x + n_y + n_z + \frac{3}{2}) \hbar \omega_0 \q... ...ar \omega_0 \qquad \Omega(N,q) = \frac{(q+3N-1)!}{q! (3N-1)!} \end{displaymath}


\begin{displaymath} S = k_B\ln \Omega \qquad \frac{1}{T} = \frac{dS}{dE} \qqua... ...0} \qquad C = \frac{1}{n} \frac{dE}{dT} \qquad E = 3Nk_B T \end{displaymath}


\begin{displaymath} \vert\vec F_C\vert = k_e {\vert q_1\vert \vert q_2\vert \ove... ...^2} \hat r_i \qquad \vert\vec E\vert = \int {k_e dq \over r^2} \end{displaymath}


\begin{displaymath} \vert\vec E\vert = 2 \pi k_e \sigma \qquad \vert\vec E\vert = k_e {qz \over (z^2 + R^2)^{3/2}} \end{displaymath}


\begin{displaymath} W \equiv \int \vec F \cdot d\vec s \qquad \Delta V \equiv {\... ...\int_A^B \vec E \cdot d\vec s \qquad V = k_e {q\over r} \qquad \end{displaymath}


\begin{displaymath} V = k_e \sum_n {q_n \over r_n} \qquad V = k_e \int {dq \over... ...partial y} \qquad E_z = - {\partial V \over \partial z} \qquad \end{displaymath}


\begin{displaymath} V = \frac{k_e Q}{\sqrt{x^2 + a^2}} \qquad I \equiv {dQ \over... ...uiv} = \sum R_i \qquad {1 \over R_{equiv}} = \sum {1\over R_i} \end{displaymath}


=100000

=2.5in The algebraic sum of the potential changes across all the elements of a closed loop is zero. The sum of the currents entering a junction is equal to the sum of the currents leaving the junction.



\begin{displaymath}\vec F_B = q \vec v \times \vec B \qquad \vert\vec F_B\vert ... ... = N_0 e^{-\lambda t} \quad t_{1/2} = \frac{\rm ln 2}{\lambda} \end{displaymath}


\begin{displaymath} y = A \cos (kx - \omega t) \quad k\lambda = 2 \pi \quad \om... ...ad B = B_m \sin (kx - \omega t) \quad I \propto {Amplitude}^2 \end{displaymath}


\begin{displaymath} I = 4 I_0 \cos^2 \left ( {\pi d \over \lambda} \sin \theta \... ... \theta \right )}{\frac{\pi a}{\lambda} \sin \theta}\right ]^2 \end{displaymath}


\begin{displaymath} \delta = d \sin \theta = m \lambda \quad \delta = a \sin \th... ...a \quad \phi = k\delta \quad \sin \theta_R = {\lambda \over a} \end{displaymath}


\begin{displaymath} \Delta t = \frac{\Delta t_p}{\sqrt{1-\frac{v^2}{c^2}}} \quad... ...^\prime = x - vt \quad y^\prime = y \quad v_x = v_x^\prime + v \end{displaymath}


\begin{displaymath} {d \over dx} x^n = nx^{n-1} \qquad \int x^n dx = {x^{n+1} \o... ...{1}{x} \qquad \frac{df(u)}{dx} = \frac{df(u)}{du}\frac{du}{dx} \end{displaymath}


\begin{displaymath} {d \over dx} (\cos ax) = -a\sin ax \qquad {d \over dx} (\sin ax) = a\cos ax \qquad \frac{d e^x}{dx} = e^x \end{displaymath}


\begin{displaymath} \ln (a\times 10^b) = \ln a + b\ln 10 \qquad e^x = 10^y \quad... ...um_i \left( x_{i}-\left\langle x\right\rangle \right)^2}{N-1}} \end{displaymath}


\begin{displaymath} x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \qquad f(x) = f(a) + ... ...rime (a) (x-a) + \frac{f^{\prime\prime}(a)}{2}(x-a)^2 + \cdots \end{displaymath}


\begin{displaymath} (a + b)^n = a^n + \frac{n}{1!}a^{n-1}b + \frac{n(n-1)}{2!}a^{n-2}b^2 + \cdots \end{displaymath}


\begin{displaymath} \int \frac{1}{\sqrt{x^2 + a^2}} dx = \ln \left [ x + \sqrt{x... ...] \qquad \int \frac{x}{\sqrt{x^2 + a^2}} dx = \sqrt{x^2 + a^2} \end{displaymath}


\begin{displaymath} \int \frac{x^2}{\sqrt{x^2 + a^2}} dx = \frac{1}{2} x \sqrt... ...rac{1}{2} a^2 \ln \left [ x + \sqrt{x^2 + a^2} \right ] \qquad \end{displaymath}


\begin{displaymath} A = 4\pi r^2 \qquad V = Ah \qquad V = {4\over 3} \pi r^3 \end{displaymath}