Physics 215 Midterm Exam

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

  1. Consider the model we used for air friction

    $\displaystyle F_f = \frac{1}{2} D \rho A v^2 $

    where $ D$ is the drag coefficient, $ \rho$ is the density of air, $ A$ is the cross sectional area of the falling object, and $ v$ is the speed. What are the units of $ D$. Why?

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

  3. Recall Hooke's Law which is represented by the differential equation

    $\displaystyle F_s = -kx$    

    where $ x$ is the distance from equilibrium and $ k$ is the spring constant. The solution is on the equation sheet. What is the `phase constant' in the solution? You may find making a sketch helpful.

    \includegraphics{f1.eps}

  4. What is torque? Be sure you explain all mathematical symbols you use in your answer.

  5. What is dynamical chaos?

  6. In class, we discussed adding a small amount of $ \rm ^{232}U$ to a uranium pit. What do we hope to gain by doing this?

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


1. (10 pts.)

The terminal speed of a skydiver in the spread-eagle position is $ \rm 160~km/hr$. In the nose-dive position, the terminal speed is $ \rm 310~km/hr$. Assuming that the drag coefficient $ D$ does not change from one position to the next, find the ratio the effective cross-sectional area $ A$ in the slower position to that in the faster position.

2. (20 pts.)

Provide an estimate to third order of the Fresnel integral.

$\displaystyle \int_0^1 \sin (x^2) dx $

3. (20 pts.)

A body oscillates with simple harmonic motion according to the equation

$\displaystyle x = (6.0~m) \cos\left ( (3\pi~rad/s)t + \pi/3~rad \right ). $

At $ t=2.0~s$ what are the displacement, the velocity, and the acceleration?

4. (20 pts.)

The child/daughter nucleus formed in radioactive decay is often radioactive itself. Let $ N_{10}$ be the initial number of parent nuclei at $ t=0$, $ N_1(t)$ be the number of parent nuclei at some later time $ t$, and $ \lambda_1$ be the decay constant. Suppose the number of child nuclei at $ t=0$ is zero and let $ N_2(t)$ be the number of child nuclei at time $ t$ and $ \lambda_2$ be the decay constant of the child. The equation for $ N_2(t)$ is the following.

$\displaystyle N_2(t) = \frac{N_{10} \lambda_1}{\lambda_1 - \lambda_2} \left ( e^{-\lambda_2 t} - e^{-\lambda_1 t} \right )$    

Where/When is the maximum value of $ N_2(t)$? The isotope $ \rm ^{211}Bi$ alpha decays to $ \rm ^{207}Tl$ with $ t_{1/2} = 2.14~min$. The child nucleus $ \rm ^{207}Tl$ then beta decays to stable $ \rm ^{207}Pb$ with a half life $ t_{1/2} = 4.77~min$. When is the number of $ \rm ^{207}Tl$ nuclei at a maximum?

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}$

$\displaystyle \vec F = m \vec a = m {d^2 \vec r \over dt^2} = {d\vec p \over dt... ... \qquad \vec F = - {1 \over 2} C \rho A v^2 \hat v \qquad \vec F = - Dv \hat v $

$\displaystyle y(t) = {1 \over 2} at^2 + v_0 t + y_0 \qquad v(t) = at + v_0 F_s = -kx \qquad x(t) = A\cos (\omega t + \phi) \qquad \omega T = 2 \pi \qquad $

$\displaystyle K = m_0 c^2 \left( {1 \over \sqrt{1-(v^2/c^2)} } - 1 \right) \qquad K = \frac{1}{2} mv^2 \qquad p = { {m_0 v} \over \sqrt{1-(v^2/c^2)} } \qquad $

$\displaystyle \vert\vec \tau \vert = I\alpha = I {d^2\theta \over dt^2} = F_{\perp} r \qquad L = {I \over mL_{cm}} \qquad I = \sum_i mr_i^2 = \int r^2 dm $

$\displaystyle \tau = - mg\sin \theta \qquad \tau = F_f L_{cm} = -D \omega L_{cm} \qquad \tau = F_D \sin (\Omega t)~ L_{cm} $

$\displaystyle f(x) = \sum_{n=0}^{\infty} {f^{(n)}(x_0) \over n!}(x - x_0)^n \qq... ...\over h} + O(h) \qquad f_0^{\prime} = { f_{n} - f_{n-1} \over h} + O(h) \qquad $

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

$\displaystyle \frac {dN}{dt}= -\lambda N \qquad N = N_0 e^{-\lambda t} $

$\displaystyle {d \over dx} x^n = nx^{n-1} \qquad \int x^n dx = {x^{n+1} \over n+1} + c \quad $

$\displaystyle \int { dx \over (x^2 + a^2 )^{3/2}} = {x \over a^2(x^2 + a^2)^{1/2}} \qquad \int {dx \over \sqrt{x^2 + a^2}} = \ln(x + \sqrt{x^2 + a^2}) $

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

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