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PHYSICS 231
Final Examination, December 22, 1997, 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

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

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

${\bf \tau = r \times F}$; 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.

a x² + b x + c=0, $x= \frac{-b \pm \sqrt{b^2 -4 a c}}{2 a}$

g=9.8m/s² = 32 ft/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) While two forces act on it, a particle is to move continuously with $\vec{v} = (3 m/s) \hat{i} -(4 m/s) \hat{j}$. One of the forces is $\vec{F}_{1} = (2 N)\hat{i} +(-6N)\hat{j}$. What is the other force? 10 pts.

2) An object is tracked by a radar station and found to have a position vector given by $\vec{r} = (3500 - 160 t) \hat{i} + 2700 \hat{j} + 300 \hat{k}$, with $\vec{r}$in meters and t in seconds. The radar station's x axis points EAST, its y axis points NORTH, and its z axis VERTICALLY UP. If the object is a 250 kg meteorological missile, what are (a) its linear momentum (4 pts), (b) its direction of motion (3 pts), and (c) the net force on it (3 pts)?

3) A student wants to determine the coefficients of static friction and kinetic friction between a box and a plank. She places the box on the plank and gradually raises one end of the plank. When the angle of inclination with the horizontal reaches 30⁰, the box starts to slip, and it slides 2.5 m down the plank in 4.0 s. What are the coefficients of static and kinetic friction? 10 pts.

4) The only force acting on a 2.0 kg body as it moves along the x axis varies as shown in the Figure below. The velocity of the body at x=0 is 4.0 m/s. (a) What is the kinetic energy of the body at x=3.0 m (4 pts)? (b) At what value of x will the body have a kinetic energy of 8.0 J (3 pts)? (c) What is the maximum kinetic energy attained by the body between x=0 and x=5.0 m (3 pts)?

5) The cable of the 4000 lb elevator shown in the Figure below snaps when the elevator is AT REST at the first floor, where the bottom is a distance d = 12 ft above a cushioning spring whose spring constant k = 10,000 lb/ft. A safety device clamps the elevator against guide rails so that a constant frictional force of 1000 lb OPPOSES the motion of the elevator. (a) Find the speed of the elevator JUST BEFORE it hits the spring. (5 pts) (b) Find the maximum distance x that the spring is compressed. (5 pts)

6) A child of mass M stands on the rim of a merry-go-round (which can rotate without friction) of radius R and rotational inertia I that IS NOT MOVING. The child throws a rock of mass m HORIZONTALLY in a direction that is TANGENT to the outer edge of the merry-go-round. The speed of the rock, relative to the ground, is v. Afterwards, what are (a) the angular speed of the merry-go-round (6 pts) and (b) the linear speed of the child (4 pts)?

7) An oscillating block-spring system has a mechanical energy of 1.0 J, an amplitude of 10.0 cm, and a MAXIMUM speed of 1.2 m/s. Find (a) the spring constant (4 pts), (b) the mass of the block (4 pts) and (c) the frequency f of the oscillation (2 pts).

8) The mass of the H₂ molecule is $3.3 \times 10^{-27} kg$. If 10²³ H₂ MOLECULES PER SECOND strike 2.0 cm² of wall AT AN ANGLE 55⁰ with the normal when moving with a speed of 10³ m/s, what pressure do they exert on the wall? 10 pts.

9) An IDEAL GAS initially at 300 K is compressed at a constant pressure of 25 N/m² from a volume of 3.0 m³ to a volume of 1.8 m³. In the process, 75 J of heat is LOST by the gas. What are (a) the change in internal energy of the gas (5 pts) and (b) the final temperature of the gas (5 pts)?

10) A pickup truck whose mass is 2200 kg is speeding along the highway at 65 miles/hour. If you could use all this kinetic energy to vaporize water ALREADY at 100⁰ C, how much water (in terms of mass) could you vaporize (8 pts)? If you had to buy this amount of energy from your local utility company at 12 cents /kW.h, how much would it cost you (2 pts)? 1 mile = 1.61 km.