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PHYSICS 231
Final Examination, December 10, 1996, 9:00am-12:00pm
Instructor: P. Q. Hung

10 points for each problem. READ the problems carefully. SHOW YOUR WORK. DO NOT JUST write the answers down. Answers without explanations will be given no credit.

Useful formulas (not necessarily applied to all problems below):

x=x₀+v_0xt+(1/2)a_xt²; y=y₀+v_0yt+(1/2)a_yt²; v_x=v_0x+a_xt; v_y=v_0y+a_yt

$v^2=v_0^2 +2 a \Delta x$;$R= v_o^2 sin(2\theta_0)/g$

$f=\mu N$; F_G= -GMm/r²; F=mv²/r, F= -kx

$W= \int F(x) dx$, $P= \frac{dW}{dt}$

$E= \frac{1}{2} \ m \ v^2 + U$;${\bf p} = m {\bf v}$; ${\bf L = r \times p}$;$\omega = \sqrt{k/m}$

${\bf \tau = r \times F}$; c_water = 4.18 kJ/kg.K; C=mc

PV = n RT; R=0.082 L.atm/mol.K; $W=nRT\ln(V_f/V_i)$;$Q=C\Delta T$; $K_{trans}=\frac{1}{2} k T$ for each degree of freedom.

L_f = 333.5 kJ/kg and $L_v = 2.257 \times 10^{3}$ J/g for water; 1 Pa = 1 N/m²; $k= 1.38 \times 10^{-23} J/K$.1 atm = 101.3 kPa.

$\int x^{n} dx = \frac{x^{n+1}}{n+1}$, $\frac{d}{dx} (sinax) = a cosax$,$\frac{d}{dx} (cosax) = -a sinax$.

$\hat{\imath} \times \hat{\jmath} = \hat{k}$ and cyclic permutations.

g=9.8m/s², $G= 6.67 \times 10^{-11} m^3/kgs^2$, 1 eV = $1.602 \times 10^{-19}$ J. $c = 3 \times 10^{8}$ m/s. 1 W = 1 J/s.

1) One mole of N₂ gas initially at a temperature of 300 K and pressure of 1 atm is compressed isothermally until its volume is reduced by a factor of 10. (a) What is the final pressure of the gas? 3 pts (b) What is the work done by the gas or on the gas? 3 pts (c) How much heat flow is transferred? 4 pts.

2) (a) At what temperature would the thermal speed of nitrogen molecules in a gas be 1% of the speed of light c? 9 pts (b) Why might it not be possible to achieve this temperature in N₂ gas? (A reasonable qualitative reason for (b) will be sufficient) 1 pt. Mass of N₂ molecule = $4.65 \times 10^{-26}$ kg. Typical molecular binding energy $\approx$ few eV.

3) A pan contains 0.5 kg of water at 25⁰ C. (a) How much heat does it require to raise the temperature of the water to 100⁰ C and vaporize it at 1 atm? 4 pts. (b) How much work is done by the water as it changes to the gaseous phase? 3 pts. (c) What is the total change in the internal nergy of the water? 3 pts.

4) A mass m= 1.5 kg is attached to an ideal spring of spring constant k= 18 N/m. All motion takes place in the (horizontal) x-direction, and the equilibrium position of the mass is defined to be at x=0 m. The mass is then displaced to x=0.25 m and released from rest. (a) Write an expression for the function x(t) that describes the subsequent motion. 4 pts. (b) At what position x does the maximum positive acceleration of the mass occur and what is the magnitude of that maximum acceleration? 4 pts. (c) What is the maximum speed of the mass? 2 pts.

5) 650 g of water at 20⁰ C is put in a pan on a 1.5 kW stove burner. How long will it take to boil the pan dry, assuming all the burner's energy goes into the water? 10 pts.

6) A cross-country skier moving at 5.0 m/s on level ground encounters a nearly frictionless downward slope 6.0 m high. On the level ground below, the snow has been worn thin, and consequently, the coefficient of friction is 0.3. After coasting down the hill, how far will the skier glide across the level stretch? 10 pts.

7) A surface is frictionless except for a region between x=1 m and x=2 m, where the coefficient of friction is given by $\mu = ax^2+ bx +c$, with a= -2m^-2, b= 6 m^-1, and c=-4. A block is sliding in the +x direction when it encounters this region. What is the minimum speed it must have to get all the way across the region? 10 pts.

8) A rock of mass 200 g is thrown with initial horizontal speed v_x = 20 m/s off a building from a height of 30 m (see figure below). Calculate the angular momentum, $\vec{L}$, of the rock about point O (the origin of the coordinate system shown in the figure) as a function of time. 10 pts.

9) A crate of mass 250 kg is loaded on the back of the truck. The coefficient of static friction between the crate and the truck bed is $\mu_{s}$. The truck decelerates suddenly, so that it stops from a speed of 26.7 m/s to a distance of 140 m. How large must $\mu_{s}$ be so that the crate does not slide forward on the truck bed? 10 pts.

10) A billiard ball with velocity $\vec{v} = (1.80 m/s) \hat{\imath}$ strikes a stationary billiard ball of the same mass. After collision, the first billiard ball has velocity $\vec{v}_{1} = (1.44 m/s) \hat{\imath} + (0.72 m/s) \hat{\jmath}$. (a) What is the velocity (this is a vector please) of the second ball? 5 pts. (b) Is the collision elastic? 5 pts.