\documentstyle[12pt]{article} \setlength{\oddsidemargin}{0in} \setlength{\evensidemargin}{0in} \setlength{\textwidth}{6.5in} \setlength{\topmargin}{-.8in} \setlength{\textheight}{10in} \pagestyle{empty} \title{} \author{} \date{} \begin{document} \begin{center} \begin{Large} {\bf Physics 101 \hspace*{10ex} Final Exam \hspace*{8ex} Dec. 6 1999} \end{Large} \end{center} \vspace*{5ex} {\bf \hspace*{50ex} Name: \underline{ \hspace*{20ex}} } \\ \begin{center} This test is administered under the rules and regulations \\ of the honor system of College of William \& Mary. \\ \end{center} {\bf \hspace*{50ex} Signature: \underline{ \hspace*{20ex}} } \\ \hspace*{28ex} {\bf Problem Session Instructor: \underline{ \hspace*{20ex}} } \\ \vspace*{2ex} \begin{center} \parbox{5.0 cm} { 1. \underline{ \hspace*{10ex}} (Carone) \\ \vspace*{1ex} 2. \underline{ \hspace*{10ex}} (Eckhause) \\ \vspace*{1ex} 3. \underline{ \hspace*{10ex}} (Schone) \\ \vspace*{1ex} 4. \underline{ \hspace*{10ex}} (Averett) \\ \vspace*{1ex} 5. \underline{ \hspace*{10ex}} (Armstrong) \\ \vspace*{1ex} 6. \underline{ \hspace*{10ex}} (Carone) \\ \vspace*{1ex} 7. \underline{ \hspace*{10ex}} (Schone) \\ \vspace*{1ex} 8. \underline{ \hspace*{10ex}} (Eckhause) \\ } \end{center} \vspace*{8ex} (equation sheet provided separately) \newpage \begin{Large}Problem 1. \end{Large} \\ A boy is fishing in a stationary (unmoored) boat. Being ecologically conscious, he throws back the 2.0 kg fish he has caught. He gives the fish an initial horizontal velocity of 5.0 m/s relative to the water. The mass of the boy and the boat together is 45 kg. The fish is a largemouth bass, not a red herring. \\ \noindent a) What is the velocity $v_b$ of the boy and boat after the fish is thrown? \\ \noindent b) How much energy did the boy have to expend in throwing the fish? \\ \newpage \begin{Large}Problem 2. \end{Large} \\ An alien spacecraft is hiding behind the moon, such that the Earth, the Moon, and the spacecraft are all in a straight line, with the Moon somewhere between the Earth and the spacecraft. Calculate the distance $r$ between the Moon and the spacecraft such that the Earth and the Moon exert equal forces on the alien. \\ \noindent Earth-Moon distance $R_{EM} = 3.84 \times 10^8 \; {\rm m}$ \\ Mass of the Earth $M_E = 5.98 \times 10^{24} \; {\rm kg}$ \\ Mass of the Moon $M_m = 7.26 \times 10^{22} \; {\rm kg}$ \\ \newpage \begin{Large} Problem 3. \end{Large} \\ A 60 kg block slides along the top of a 100 kg block with an acceleration $a = 3\; {\rm m/s}^2$ when a horizontal force of 280 N is applied. The 100 kg block sits on a horizontal frictionless surface, but there is friction between the two blocks. \\ \noindent a) Find the coefficient of kinetic friction between the blocks. \\ \noindent b) Find the acceleration (magnitude and direction) of the 100 kg block (when it is still in contact with the other block). \newpage \begin{Large}Problem 4. \end{Large} \\ A mass $m = 2.0$ kg is sitting on a massless board \\ a distance $D = 1.5 $ m to the left of a support. \\ The other end of the board, which is at a distance $L = 2.5 $ m \\ from the support, is tied to the floor by a (vertical) string. \\ \noindent a) What is the tension in the string? \\ \noindent b) What is the force that the support exerts on the board? \\ \noindent c) Someone cuts the string. What is the magnitude and direction of the {\em initial} angular acceleration of the board ? \\ \newpage \begin{Large}Problem 5. \end{Large} \\ A William \& Mary soccer player kicks a soccer ball (mass = 0.380 kg) with an initial speed of 15 m/s at an angle of 20$^{\circ}$ to the horizontal when she was 11 m from the front of the goal. The height of the top of the goal is 2.5 m from the ground. Does the ball go in the goal, or does it fly over the top? Ignore air resistance. Show your work! \newpage \begin{Large}Problem 6. \end{Large} \\ A cylinder is supported in water ($\rho_{\rm water} = 1 {\rm g/cm^3}$) \\ by a string that will break if its tension is greater than \\ $T = 45$ N. The cylinder has a cross sectional area $A = 120 {\rm cm}^2$, \\ a length $L= 20$ cm, and a mass of $m = 5.5$ kg. \\ \noindent a) What is the tension in the string while the cylinder is completely submerged? \\ \noindent b) What fraction of the length of the cylinder can be pulled out of the water before the string breaks ? \newpage \begin{Large}Problem 7. \end{Large} \\ A string of mass $m = 2 \times 10^{-3}$ kg and a length $L = 3$ m oscillates in a standing transverse wave of one and a half wavelengths, at a frequency $f = 60$ Hz. \\ \noindent a) What is the tension in the string? \\ \noindent b) What would be the velocity of a travelling wave in the string? \\ \newpage \begin{Large}Problem 8. \end{Large} \\ Consider the vectors $\vec{A} = 3\vec{i} + 4\vec{j} - 2\vec{k}$ and $\vec{B} = 2\vec{i} - 5 \vec{j}$. \\ \noindent a) What are the magnitudes of $\vec{A}$ and $\vec{B}$? \noindent b) What is the angle between $\vec{B}$ and the $x$-axis? \noindent c) What is $\vec{A} - 2\vec{B}$? \noindent d) What is $\vec{A}\cdot\vec{B}$? \noindent e) What is $\vec{A} \times \vec{B}$? \newpage \begin{large} {\bf {Possibly useful relations:}} \\ \end{large} \begin{table}[hb] {\protect\setlength{\baselineskip} {4ex} } %\vspace{1mm} \begin{tabular}{lll} $x = x_0 + v_0 t + \frac{1}{2}a t^2$ & $v = v_0 + a t $ & $v^2 = v_0^2 + 2 a \Delta x $ \\ $\vec{v}_{\rm average} = \Delta \vec{x} / \Delta t$ & $\vec{a}_{\rm average} = \Delta \vec{v} / \Delta t$ & $\vec{F}_{AB} = - \vec{F}_{BA}$ \\ $\vec{a} = d\vec{v}/dt$ & $\vec{v} = d\vec{x}/dt $ & $\Sigma\vec{F} = m \vec{a}$ \\ $a_c = \frac{v^2}{r}$ & $\vec{W} = m \vec{g}$ & $F_x = -k \Delta x$ \\ $\vec{v}_{B} = \vec{v}_{A} + \vec{v}_{AB}$ & $R = \frac{v_0^2}{g} \sin{2 \theta} $ & $ v = \frac{2\pi r}{T}$ \\ %$\cos{\theta}$ = adjacent/hypotenuse & \\ %$\sin{\theta}$ = opposite/hypotenuse & \\ %$\tan{\theta}= \sin{\theta}/\cos{\theta}$ & \\ $x = \frac{-b \pm \sqrt{b^2 - 4 a c}}{2a}$ & $0 \leq f_s \leq \mu_s F_n$ & $f_k = \mu_k F_n$ \\ $W = F_x \Delta x$ & $W_{\rm net} = \Delta K$ & $K = \frac{1}{2} m v^2$ \\ $W = \int F_r dr = \int \vec{F}\cdot d\vec{r}$ & $E = K + U$ & $\vec{A}\cdot\vec{B}= A B \cos{\phi}$ \\ $\Delta U = -W = -\int \vec{F}\cdot d\vec{r}$ & $U = mgy + U_0$ & $U = \frac{1}{2} k x^2$ \\ $W_{nc} = \Delta E$ & $P = \frac{dW}{dt} = \vec{F}\cdot \vec{v}$ & $ F(x) = -dU/dx $ \\ $M\vec{R}_{CM} = \Sigma m_i \vec{r}_i$ & $M\vec{R}_{CM} = \int \vec{r} dm$ & \\ $\vec{p} = m \vec{v}$ & $\vec{F} = \frac{d\vec{P}}{dt}$ & $ \vec{J} = \int \vec{F}dt = \vec{F}_{av}\Delta t = \Delta\vec{p}$ \\ $\omega = \frac{d\theta}{dt}$ & $\alpha = \frac{d\omega}{dt}$ & $ v = r \omega$ \\ $a_T = r \alpha$ & $\omega = \omega_0 + \alpha t$ & $ \theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$ \\ $\omega^2 =\omega_0^2 + 2\alpha(\theta - \theta_0)$ & $\vec{\tau} = I \vec{\alpha} $ & I = $\Sigma m_i r_i^2 $ \\ $K = \frac{1}{2}I \omega^2$ & $P =\tau \omega$ & $I = I_{\rm CM} + Md^2$ \\ $ \vec{\tau} = \frac{d\vec{L}}{dt}$ & $ V_{CM} = R \omega$ & $ A_{CM} = R \alpha$ \\ $\vec{A} \times \vec{B} = (AB \sin\theta ) \hat{n}$ & $\vec{\tau} = \vec{r} \times \vec{F}$ & $\vec{L} = \vec{r} \times \vec{p}$ \\ $\vec{L} = I \vec{\omega}$ & $X_{CG}W = \Sigma w_i x_i $& $ T^2 = \frac{4 \pi^2}{GM_s} R^3 $ \\ $U(r) = + \frac{G M_E m}{R_E} - \frac{G M_E m}{r}$ & $U(r) = - \frac{G M_E m}{r} $ & $v_E = \sqrt{2 G M_E/R_E}$ \\ $\vec{F}_{12} = -\frac{G m_1 m_2}{r_{12}^2} \hat{r}_{12}$ & $\tau = rFsin\theta$ & $k = \frac{2\pi}{\lambda}$\\ $ f = \frac{\omega}{2 \pi} $ & $ T = 1/f $ & $\omega = \sqrt{k/m}$ \\ $T = 2\pi \sqrt{L/g}$ & $x = A \cos(\omega t + \delta)$ & $E_{Total} = \frac{1}{2} k A^2 $ \\ $ x = A_0e^{-(b/2m)t}\cos(\omega't + \delta)$ & $\tau = m/b$ & $ Q = \omega_0 \tau = 2\pi \frac{\tau}{T}$ \\ $ \omega' = \sqrt{\omega_0^2 - \frac{b^2}{4m^2}}$ & $Q = 2\pi \frac{E}{\Delta E}$ & \\ $v = \sqrt{T/\mu}$ & $v = \lambda f$ & $y(x,t) = A \sin (kx - \omega t)$ \\ $P = \frac{1}{2} \mu \omega^2 A^2 v$ & $y_n(x,t) = A_n \cos (\omega_n t)(\sin k_n x)$ & $p_0 = \rho \omega v s_0$ \\ $v = \sqrt{B/\rho}$ & $v = \sqrt{\frac{\gamma R T}{M}} $ & $ I = P_{av}/4\pi r^2$ \\ $\beta = 10 \; {\rm log} (I/I_0)$ & $I_0 = 10^{-12}$W/m$^2$ & $f = f_0(\frac{v \pm v_r}{v \pm v_s})$ \\ $\rho = M/V$ & $P = F/A$ & $P = P_0 + \rho g y$ \\ $F_{\rm Bouy.} = W_{\rm disp.}$ & $v_1 A_1 = v_2 A_2$ & $P + \frac{1}{2}\rho v^2 + \rho gy = {\rm const.}$ \\ & & \\ G = $6.67\times10^{-11} \; {\rm N m}^2/{\rm kg}^2$ & $\rho_{H_2O} = 10^3 \; {\rm kg}/{\rm m}^3$ & g = - 9.8 m/s$^2$ \\ \end{tabular} \end{table} %\end{large} \end{document}