Physics 132-1 Final Exam

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Questions (2 pts. apiece) Answer in complete, well-written sentences WITHIN the spaces provided.

1. Consider the twins paradox. As the spacefaring twin's craft recedes from the Earth it is moving at a constant speed. Since no inertial frame can be considered `better' than any other there is nothing physically inconsistent with the view that the spacefaring twin is observing Earth recede from her at a constant velocity. Hence, the spacefaring twin will observe clocks on the Earth to move slowly and the Earthbound twin will age at a slower rate than the spacefaring one. Is this reasoning flawed? How?

2. What would you see in a mirror if you carried it in your hands and ran at (or near) the speed of light?

3. How could you measure absolute zero?

4. In our model of an ideal gas we assumed the collisions between the molecules were elastic. Is this a reasonable assumption? Why? What would happen if it were not true?

5. Atoms and molecules have equal amounts of positive and negative charges so they are electrically neutral. We often treat them as if they had no electrical field at all. Recall our assumptions in the kinetic theory of gases. If this assumption valid? Explain.

6. How does a mass spectrometer work?

7. What is the ultimate source of magnetic fields?

8. Consider the distribution of charge shown below. What do the equipotentials lines look like in the region between the plates? A sketch might be helpful here. Explain your reasoning.

   

   
   
   

9. How does radiocarbon dating work?

10. Is light a particle or a wave? What is your evidence?

11. Consider the intensity pattern shown below for two, identical, narrow slits. How would you determine the separation of the slits?

    
    

    

    
    

12. Radionuclides `die out' by decaying exponentially. Batteries, stars, and even people `die out'. Do theses last object decay exponentially? Explain.

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

1.

8 pts.

An astronaut is traveling in a space vehicle that has a speed $v = 0.40c$ relative to the Earth. The astronaut measures her pulse rate at 72 beats/min. Signals generated by the astronaut's pulse are radioed to Earth when the vehicle is moving in a direction perpendicular to a line that connects the vehicle with an observer on the Earth. What pulse rate does the Earth observer measure?

2.	8 pts.	A Klingon spaceship moves away from the Earth at a speed $v = 0.800c$ (see figure). The starship |it Enterprise pursues at a speed $u = 0.900c$ relative to the Earth. Observers on the Earth see the Enterprise overtaking the Klingon ship at a relative speed of $0.100c$. With what speed is the Enterprise overtaking the Klingon ship as seen by the crew of the Enterprise?
3.	8 pts.	When an uncharged conducting sphere of radius $a$ is placed at the origin of an $xyz$ coordinate system that lies in an initially uniform electric field $\vec E = E_0 \hat k$, the resulting electric potential is $$V(x, y, z) = V_0 - E_0 z + \frac{E_o a^3 x}{(x^2 + y^2 + z^2)^{3/2}}$$ for points outside the sphere, where $V_0$ is the constant electric potential on the conductor. Use this equation to determine the $x$, $y$, and $z$ components of the resulting electric field.
4.	8 pts.	Two moles of an ideal monatomic gas is at an inital temperature $T_0 = 350~K$. The gas undergoes an isovolumetric (constant volume) process acquiring 600 Joules of energy from heat. What is the new temperature of the gas?

5.	10 pts.	A heart surgeon monitors the flow of blood through an artery using an electromagnetic flowmeter like the one shown in the figure. The device takes advantage of the presence of ions and electrons in blood as it flows between the poles of the magnet. Electrodes $A$ and $B$ make contact with the outer surface of the artery. The inner diameter of the artery is $d = 2.5\times 10^{-3}~m$. The magnetic field strength is $B = 0.05~T$ and a voltage difference $V = 1.0 \times 10^{-4}~V$ is measured. What is the speed of the blood? Which electrode is positive? Assume the electric field in the blood is uniform and there are equal amounts of positive and negative charge.
6.	10 pts.	One of the dangers of a nuclear explosion is radioactivity from strontium $\rm {^{90}Sr}$ which decays with a half-life $\tau = 29~yrs$. Because it is chemically similar to calcium it has the nasty habit of showing up in cow's milk. The cow's eat grass tainted with the $\rm {^{90}Sr}$ and it replaces calcium in their milk. Eventually some of the $\rm {^{90}Sr}$ ends up in the bones of whoever drinks the milk (like babies and small children). The energetic electrons emitted by the beta decay of $\rm {^{90}Sr}$ damage the bone marrow and suppress the production of red blood cells. A 250 kiloton bomb produces about $m= 0.10~kg$ of $\rm {^{90}Sr}$. Such a bomb releases about 200 times the energy of the Hiroshima bomb which killed 66,000 and injured another 69,000. We have about 6,000 nuclear bombs of this size or bigger mounted on long-range missiles. If the fallout is spread uniformly over an area $A_0 = 5.0\times 10^8~m^2$, then what area $A_1$ would receive enough radioactivity to produce $74,000 ~ counts/s$. This disintegration rate is the allowed bone burden for one person.

Constants and Equations

Coulomb's Law constant ( $k_e = {1 \over 4\pi\epsilon_0}$)	$8.99\times 10^{9} {N-m^2 \over C^2}$	Electron mass	$9.11\times 10^{-31}~kg$
Elementary charge ($e$)	$1.6\times 10^{-19}~C$	Proton/Neutron mass	$1.67\times 10^{-27}~kg$
Permittivity constant ($\epsilon_0$)	$8.85\times 10^{-12} {kg^2\over N-m^2}$	$\rm 1.0~eV$	$\rm 1.6\times 10^{-19}~J$
$1 ~ u$	$1.67\times 10^{-27}~kg$	$k_B$	$1.38\times 10^{-23} ~ J/K$
$c$	$3.0\times 10^{8}~m/s$	Avogadro's number	$6.022\times 10^{22}$

$${d \over dx} x^n = nx^{n-1} \qquad \int x^n dx = {x^{n+1} \o... ...um_i \left( x_{i}-\left\langle x\right\rangle \right)^2}{N-1}}$$

$$\int { dx \over (x^2 + a^2 )^{3/2}} = {x \over a^2(x^2 + a^2... ... \int {dx \over \sqrt{x^2 + a^2}} = \ln(x + \sqrt{x^2 + a^2})$$

$$\int {x~ dx \over (x^2 + a^2 )^{3/2}} = - {1 \over (x^2 + a^... ...ln \vert x\vert \int {x dx \over x+d} = x - d\ln (x+d) \qquad$$

$${d f(u) \over dx} = {df\over du}{du \over dx} \qquad {d \ \o... ...(\cos ax) = -a\sin ax \qquad {d \over dx} (\sin ax) = a\cos ax$$

$$A = 4\pi r^2 \qquad V = Ah \qquad V = {4\over 3} \pi r^3\qquad x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

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

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

$$\vec F = m \vec a = {d\vec p \over dt} \quad \langle \vec F ... ...KE + PE\quad \vec p = m \vec v \quad \vec p_0 = \vec p_1 \quad$$

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

$$v_{rms} = \sqrt{\overline {v^2} } \qquad C_{V} = {f\over 2} ... ... 2} Nk_BT \qquad f \equiv \mbox{ number of degrees of freedom}$$

$$\vert\vec F_G\vert = G \frac{m_1 m_2}{r^2} \qquad \vert\vec ... ... r_i^2} \hat r_i \qquad \vert\vec E\vert = \int {dq \over r^2}$$

$$\vert\vec E\vert = k {Q \over r^2} \qquad \vert\vec E\vert =... ...n_0} \qquad \vert\vec E\vert = k {qz \over (z^2 + R^2)^{3/2}}$$

$$W \equiv \int \vec F \cdot d\vec s \qquad \Delta V \equiv {\... ...k \sum_n {q_n \over r_n} \qquad V = k \int {dq \over r} \qquad$$

$$E_x = - {\partial V \over \partial x} \quad E_y = - {\partia... ...quiv} = \sum R_i \quad {1 \over R_{equiv}} = \sum {1\over R_i}$$

=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.

$$x = \frac{a}{2} t^2 + v_0 t + x_0 \qquad v = at + v_0 \qquad F_g = mg \qquad a_c = \frac{v^2}{r}\qquad$$

$$\vec F = m \vec a \quad \vec F_B = q \vec v \times \vec B \q... ... v \vert\vec B\vert \sin \theta \quad \vec F_E = q\vec E \quad$$

$${d N \over d t} = - \lambda t \quad N = N_0 e^{-\lambda t} \quad t_{1/2} = \frac{\rm ln 2}{\lambda}$$

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

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

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

$$\Delta t = {\Delta t_p \over \sqrt {1 - {v^2 \over c^2}}} \q... ...^\prime = x - vt \quad y^\prime = y \quad v_i^\prime = v_i - v$$