101f06 - Physics 171.101 Final Exam Answers May 15, 2006 '...

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Unformatted text preview: Physics 171.101 Final Exam Answers May 15, 2006 ' Prof. Barnett Take g = 10 m/sec2, sin(37°) = 0.6, 0°C: 273°K, R = 8 Joule/(mole—°K) Use a pen. You may use material written on four 3” X 5” index cards. 1, Two vectors are given by 171 2 (4? + 23’ + 31%) and 172 = (53 — 143). A) (10 pts) If the vectors are two forces, ['71 2 (4% + 23' + 31%) N, and F; 2 (53 —A143) N, acting on a 5 kg mass, what is the mass’s acceleration, a? (1.82' — 243' + 0.6k)m/sgi B) (10 pts) If instead the first vector is the total force Flow, 2 (4'? —|— 23' + 3%) N acting on an object and the second vector is the distance d 2 (Si — 143') meters moved by the object, what is the work, W, done by Fm“; during this motion? —8.0 Joules C) (10 pts) Suppose instead that the first vector is the location, B : (43+23 +31%) meters, of a 5 kg mass relative to an origin and the second vector is the linear velocity of the mass, 17 : (5i — 143') m/sec. What would the angular momentum E be for the object relative to the origin? (210; + 753 — 3301;)199 - m2/sec D) (10 pts) Suppose finally that the two vectors represent the velocities of an incom- pressible fluid at two points as the fluid travels through a pipe. If the cross sectional area of the pipe at the location #1, where 171 2 (42 + 25' + 31%) m/sec is the fluid velocity, is 1.0 cm2 what is the cross sectional area of the pipe at location #2 where 172 2 (5i — 143) m/sec is the fluid velocity? 0.3622 cm2. 2. George Washington is 50 meters above the Potomac George 25 III/886 River. He throws a silver dollar across the river, i.e. per— ‘5' ’2 370 pendicular to the shore. It leaves his hand at t = 0 with - I: “" '— “‘ a speed of 25 m/sec at an angle of 37" above the hori— zontal. Benedict Arnold is in a small boat on the river 50 m . and is drifting down river with the current which is 3 Benedict m / sec. Benedict catches the dollar and sails to England 1 with it. Use a coordinate system where i is horizontal and perpendicular to the shore, points up and 1:: is the direction of the river’s current parallel to the shore. A) (10 pts) When does Benedict catch the coin? 5.0 sec. B) (10 pts) What is the coin’s minimum speed during its flight? 20 m/sec. C) (10 pts) How far is Benedict from the shore? 100 meters D) (10 pts) What is the coin’s velocity, Va, according to George when Benedict catches it? (202' — 35j) m/sec E) (10 pts) What is the coin’s velocity, 1713,“ when Benedict catches it as measured in Benedict’s coordinate system? (201' — 35j — 3k) m/sec 3. At t = 0 a pulse on a string has a shape given by the equation 2.00m3 W“) = Both :c and 7,0 are measured in meters. The pulse has a Speed of 5m/s in the —i direction, A) (5 pts) Sketch 1/)(Lr, 0) in the region from *20m < a: < 20m. B) (10 pts) Draw a sketch of the pulse 1 second later, i.e. «Mr, 1), from —20m < a: < 20m. C) (15 pts) Write down the functional form for the pulse at an arbitrary time, i.e. rb(a:, t). 2.007723 W“) = 4. A cylinder is rolled without slipping from rest by a constant force of magnitude F0 in the horizontal direc- tion applied by a string wrapped around the cylinder. [disk 2 A) (5 pts) If the point P on the horizontal section of the string moves a distance a: relative to the ground, how far does the center of the cylinder move? x/ 2 B) (10 pts) If point P moves by a distance 51:, how much work is done by the force? xF ) C (10 pts) If the center of the cylinder moves with a speed S what is the kinetic energy of the center of mass and What is the kinetic energy of the cylinder about the center of mass? KECM = 0.5m5'2, KEaCM = 0.25mS2 D) (15 pts) When the point P has moved by a distance :12, what is the speed 5 of the center of mass of the cylinder? 3 : (/4xF/3m . ‘ ' ‘ '- k = 500 N/m 6 2 5. A pan of negligible mass hangs from a spring with spring ‘ 0 '00 kg constant k A: 500 N/m. Define the pan’s position to be y y = 0 with j pointing upward. g : 10m/sec2. 0.5 In A) (10 pts) A 2.00 kg mass is placed slowly on the pan such that it moves to another, stationary, equilibrium point. What is the location of the new equilibrium point? —4 cm B) (10 pts) If instead of being slowly placed on the pan the mass is dropped onto the pan from a height of 0.5 meters, what are the energy and momentum of the mass at the instant the mass first touches on the pan? ME = 10 J, P = 43.3253 kg - m/sec. C) (15 pts) In the situation where the mass was dropped, the mass sticks to the pan (conserving mechanical energy) and the pan then oscillates up and down. At What values of y will the pan and mass be instantaneously stationary? 0.165 m & —.244 m. D) (15 pts) What is the maximum speed of the mass during the fall and oscillation process? 3.225 m/sec 6. A Sphere has a radius of 1.0 cm at 0° C. A piece or flat metal has a hole with a radius of 0.99990 cm at 0° C. The coefficients of thermal expansion are asphm = 1.0 x lOfiG/K and 01mm, = 20.0 x 10‘6/K. A) (25 pts) At what temperature will the sphere just pass through the hole? 5.260 C 7. Two stringed instruments in an orchestra have string lengths of 1 meter and 2 meters respectively. The strings for the two instruments are made of the same material, but the diameter of a string on the larger instrument is twice that on the shorter instrument. Suppose the fundamental (first harmonic) frequency of a string on the smaller instrument is the same as the fourth harmonic of a string on the larger instrument. A) (10 pts) What is the ratio of the wave velocity on the larger instrument to that on the smaller instrument? 0.5 B) (10 pts) What is the ratio of the mass per length of the string in larger instrument to that of the smaller instrument? 4 C) (10 pts) What is the ratio of the string tension in the larger instrument to that in the smaller instrument? 1 8. Suppose the wind is blowing at 30.0 m/sec, and the temperature is 27" C. Consider 1 kg of the air, with the effective molar mass of air being 30 g/mole. NA = 6.02 x 1023 mol”1 A) (10 pts) How many Joules of energy does the 1.0 kg of air have due to the winds motion? 450 J B) (15 pts) How many Joules of energy does the 1.0 kg of air have due to the internal energy of the translational motion (ignore rotations and vibrations) of the molecules? 1.20 x 105 J P 2 9. Two moles of an ideal monatomic gas are taken around ‘ the adjacent cycle. The initial pressure is one atmo~ sphere, 105 Pascals, and the initial volume is 20.0 liters, 105 P 2 x 10‘2m3. The three processes are 1) the pressure is proportional to the volume until both double, 2) isochoric until the pressure drops to the original value, and 3) iso— I baric until the volume returns to the original value. 2 X 10—2m3 \/ A) (10 pts) What are the maximum and minimum temperatures of the gas during the cycle? 5000 K, 1250 K B) (10 pts) During the first process, going from 1 to 2, what is the heat coming into the gas, the work done by the gas and the change in the internal energy of the gas? AU=9><103 J, W=3><103 J, Q=12X103J C) (10 pts) During the second process, going from 2 to 3, what is the heat coming into the gas, the work done by the gas and the change in the internal energy of the gas? AU2—6x103 J, W20, Q=—6><103 J D) (10 pts) During the third process, going from 3 t0 1, what is the heat coming into the gas, the work done by the gas and the change in the internal energy of the gas? AU=—3><103 J, W:—2x103 J, Q2—5x103 J E) (10 pts) What is the efficiency of this engine? 8.33% F) (10 pts) What is the entropy change of the gas during one complete cycle? 0 ...
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101f06 - Physics 171.101 Final Exam Answers May 15, 2006 '...

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