Physics 102-1 Test 1


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Questions (8 pts. apiece) Answer in complete, well-written sentences WITHIN the spaces provided.

  1. Consider a chunk of ice that that is below its freezing point and you start to heat it. What is the relationship between temperature and added heat after the ice has completely melted, but before the water begins to boil?

  2. A known mass of warm water $m_w$ is placed in the calorimeter cup of mass $m_c$ and their temperature $T_1$ recorded. A known mass of ice $m_i$ at $T_2 = 0 ^{\circ } C$ (with no water) is added to the water and allowed to melt. The final temperature of the mixture after the ice has melted is $T_3$. What is the complete heat equation in terms of $m_w$, $m_c$, $m_i$, $T_1$, $T_2$, $T_3$ and any known specific heats?

  3. Consider an ideal gas. If the collisions of the particles of the gas with the wall perpendicular to the $x$ direction are elastic, show that the average force exerted on that wall for each collision is \( \langle F_{x} \rangle =2m\frac{v_{x}}{\Delta t_{x}} \)where $m$ is the mass of one of the particles and \( \Delta t_{x} \) is the mean time interval between collisions with the wall. Start from Newton's Second Law. Clearly show all steps.

  4. We showed that the change in internal energy of a gas is
    \begin{displaymath} \Delta E_{int} = \frac{3}{2}N k_B \Delta T \qquad . \end{displaymath} (1)

    How is $\Delta E_{int}$ related to the molar specific heat at constant volume $C_V$? Using your answer and the equation above, show how $C_V$ is related to Avogadro's number $N_A$ and $k_B$.

  5. Consider the measurements shown in the table below for a test of the impulse momentum theorem in one dimension where $I = \Delta p$. The average impulse is $\langle I \rangle = 0.32\pm 0.04 ~kg-m/s$. The average momentum change is $\langle \Delta p \rangle =0.27\pm 0.04 ~kg-m/s$. Do these data confirm the impulse-momentum theorem? Explain.

    Impulse (kg-m/s) $\Delta p$ (kg-m/s)
    0.31 0.32
    0.25 0.24
    0.34 0.30
    0.33 0.26




  6. Recall the demonstration where we poured liquid nitrogen on a balloon. The balloon shrank down so that is was essentially flat and after we ran out of liquid nitrogen the balloon warmed up and the volume gradually expanded until it was the same size before we poured nitrogen on it. Consider the work down by the gas in the balloon when it was warming up. Was the work positive, negative, or zero? Explain. If the work was not zero, did this effect increase or decrease the internal energy of the gas in the balloon?








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

1. 15 pts. What is the multiplicity of an Einstein solid with $N = 1$ and $E_{int} = 5\hbar\omega_0$? List all the microstates.

2. 17 pts. The circular, $m=40-kg$, lead piston shown in the figure floats on $n=0.1~mole$ of compressed air. The diameter of the piston is $d= 0.2~m$. What is the piston height $h$ if $\rm T=20^\circ C$?



\includegraphics[height=2.5in]{f2.eps}

3. 20 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.

  1. What is the initial internal energy of the air in the room?

  2. 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.

Physics 132-2 Equations and Constants


$T_{boiling}$ ($\rm N_2$) $77 K$ $T_{freezing}$ ($\rm N_2$) $63 K$
$T_{boiling}$ (water) $373 K$ or $100^\circ\rm C$ $T_{freezing}$ (water) $273 K$ or $0^\circ\rm C$
$L_v$(water) $2.26\times 10^6 J/kg$ $L_f$ (water) $3.33\times 10^5 J/kg$
$c$ (water) $4.19\times 10^3 J/kg-K$ $c$ (steam) $0.69 J/kg-K$
$\rho$ (water) $1.0\times 10^3 kg/m^3$ $P_{atm}$ $1.05\times 10^5 N/m^2$
$k_B$ $1.38\times 10^{-23} J/K$ proton/neutron mass $1.67\times 10^{-27} kg$
$R$ $8.31J/K-mole$ $g$ $9.8 m/s^2$
$0 K$ $\rm -273^\circ C$ $1  u$ $1.67\times 10^{-27} kg$
Gravitation constant $6.67 \times 10^{-11} N-m^2/kg^2$ Earth's radius $6.37\times 10^6 m$



\begin{displaymath} \vec F = m \vec a = {d\vec p \over dt} \quad \vec a = \frac{... ... ME_0 = ME_1 \quad \vec p = m \vec v \quad \vec p_0 = \vec p_1 \end{displaymath}


\begin{displaymath} \frac{dx}{dt} = \lim_{\Delta t \rightarrow 0} \frac{x(t+\Del... ...y(t) = -\frac{g}{2}t^2 + v_0t \qquad v_y(t) = -gt + v_0 \qquad \end{displaymath}


\begin{displaymath} Q = C\Delta 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 = {\rm force} \times {\rm dist... ...\langle \vec F \rangle = {\Delta \vec p \over \Delta t} \qquad \end{displaymath}


\begin{displaymath} P = {\vert\vec F\vert \over A} \qquad PV = Nk_B T = nRT \qquad \end{displaymath}


\begin{displaymath} \vec I = \int \vec F dt = \langle \vec F \rangle \Delta t ... ...er 2}m \langle {v^2} \rangle = {1 \over 2}m v_{rms}^2 \qquad \end{displaymath}


\begin{displaymath} P = {2 \over 3} {N\over V} \langle E_{kin} \rangle \qquad \l... ...qquad E_{int} = N ~ \langle E_{kin} \rangle = {3\over 2} Nk_BT \end{displaymath}


\begin{displaymath} v_{rms} = \sqrt{\langle {v^2} \rangle } \qquad C_{V} = {f\ov... ...\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) \hbar \omega_0 \qquad E = \su... ...ar \omega_0 \qquad \Omega(N,q) = \frac{(q+3N-1)!}{q! (3N-1)!} \end{displaymath}


\begin{displaymath} \langle x\rangle = \frac{1}{N}\sum_i x_{i} \qquad \sigma = \... ...um_i \left( x_{i}-\left\langle x\right\rangle \right)^2}{N-1}} \end{displaymath}


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