Physics 215 Midterm Exam, Spring, 2008

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

1. Which numerical method is better for calculating a first derivative, the forward difference formula or the centered formula? Why?

2. In lab, we numerically solved Newton's Second Law for a falling object (Lieutenant Chisov) in the form of a first-order ordinary differential equation

   $$m\frac{dv}{dt} = -mg - \frac{1}{2}\rho C A v^2$$

   
   
   where $\rho$ is the density of an object, $A$ is the cross-sectional area, $C$ is the drag coefficient, $g$ is the acceleration of gravity, and $m$ is the mass. How do we use this solution to find how long the poor Lieutenant fell?

3. Recall the differential equation for a physical pendulum which exhibited dynamical chaos

   $$I \ddot \theta = -mgL_{cm}\sin\theta - DL_{cm} \dot \theta + F_D L_{cm} \sin\Omega t$$

   
   
   where $L_{cm}$ is the distance from the origin to the center-of-mass of the pendulum, $I$ is the moment of inertia, $m$ is the mass, $g$ is the acceleration of gravity, $\theta$ is the angular position of the pendulum, $D$ is the drag coefficient, $F_D$ is the amplitude of the driving force, and $\Omega$ is the angular frequency of the driving force. How would you extract the Poincare section from the solution to this differential equation? What parameters in the differential equation might be useful?

4. What is the differential area in spherical coordinates that we used to develop an algorithm for choosing a random direction? Use the figure below to clearly define the angles you are using.

   

5. We examined the problem of nuclear smuggling when we studied self-attenuation of radiation. Why is this a problem especially in Russia?

6. Consider the slab of material (dark green) shown in the figure which has density $n(x)$ and flux $\vec J = J_x(x)\: \hat {i \, }$ passing through it. The width of the slab is $dx$ and the cross-sectional area is $A$ . How is the amount of material in the slab related to $n(x)$ ? How would you calculate the rate of change of the amount of material in the slab in terms of $J_x(x)$ and parameters like $A$ and $dx$ ?

   

Problems. Work your solutions out on a separate piece of paper. Clearly show all work for full credit.

1. (10 pts.)	Consider the Taylor series expansion of the exponential function $e^x$ about the origin. $$e^x = \sum_{n=0}^\infty \frac{1}{n!} x^n$$ Show the expression above is correct.
2. (15 pts.)	The variance $\sigma^2$ is the square of the standard deviation of a statistical distribution. It is defined as $$\sigma^2 =\langle ( x - \langle x \rangle )^2 \rangle$$ where the brackets mean the average value of the quantity. Show the following. $$\sigma^2 = \langle x^2 \rangle - \langle x \rangle^2$$ Be sure to justify your steps.
3. (20 pts.)	Consider the one-dimensional diffusion equation corresponding to particle in a long pipe of length $L$ $$\frac{\partial n(x,t)}{\partial t} = D\frac{\partial^2 n}{\partial x^2} + Cn$$ where $n(x,t)$ is the particle density, $D$ is the self-diffusion coefficient, and $C$ is the creation rate. What is the dispersion relationship? The solution to the diffusion equation can be found in the equation sheet.
4. (25 pts.)	Consider a baseball struck by Manny Ramirez and subject to a friction force of the form $\vec F_f = -\frac{1}{2}\rho C A v^2 \hat v$ where $\rho$ is the density, $A$ is the cross-sectional area, $C$ is the drag coefficient, $v$ is the velocity, and $\hat v$ is a unit vector in the direction of the velocity. What are the components of the total vector force on the object? Express your result from part 1 as a set of first-order, linear, ordinary differential equations where the components of the velocity vector are functions of the time $t$ . Be sure to express your answer in terms of the velocity components and any necessary constants. Generate an algorithm to solve the equations from part 2 using the two-point formula for $\vec v$ .

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	$6.022\times 10^{22}$

$$\vec F = m \vec a = m {d^2 \vec r \over dt^2} = {d\vec p \over dt... ...\hat v \quad y(t) = {1 \over 2} at^2 + v_0 t + y_0 \quad v(t) = at + v_0 \quad$$

$$F_s = -kx \quad x(t) = A\cos (\omega t + \phi) \quad \omega T = 2... ...-(v^2/c^2)} } - 1 \right) \quad p = { {m_0 v} \over \sqrt{1-(v^2/c^2)} } \quad$$

$$K = \frac{1}{2} mv^2 \quad \vert\vec \tau \vert = I\alpha = I {d^... ... \frac{I}{mL_{cm}} \quad E = - \frac{GMm}{r} \quad T^2 = \frac{4\pi^2}{GM}a^3$$

$$\tau_g = - mg\sin \theta \quad \tau_f = F_f L_{cm} = -D \omega L_... ...{cm} \quad f(x) = \sum_{n=0}^{\infty} {f^{(n)}(x_0) \over n!}(x - x_0)^n \quad$$

$$f_0^{\prime} = { f_{n+1} - f_n \over h} + O(h) \quad f_0^{\prime}... ...^2) \quad f_0^{\prime\prime} = { f_{n+1} - 2f_n + f_{n-1} \over h^2 } + O(h^2)$$

$$\frac {dN}{dt}= -\lambda N \quad N = N_0 e^{-\lambda t} \quad \l... ...artial^2 n}{\partial x^2} + Cn \quad n(x,t) \rightarrow e^{\pm i(kx-\omega t)}$$

$$e^{ix} = \cos x + i\sin x \quad {d \over dx} e^{ax} = a e ^{ax} \... ...dx = {x^{n+1} \over n+1} + c \quad \int {dx \over x} = \ln \vert x\vert \quad$$

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

$$\int {x~ dx \over (x^2 + a^2 )^{3/2}} = - {1 \over (x^2 + a^2)^{1/2}} \quad$$

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