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2006f_f - Question 1(10 points A spherical shell of mass M...

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Unformatted text preview: Question 1 (10 points) A spherical shell of mass M and radius R rolls smoothly down an incline. The rotational inertia of 2 the spherical shell about any diameter is I com 2 EMRZ. Circle the correct answer to the questions below. (a) (3 pts) The ratio between translational and rotational kinetic energy, Ktran, is (i) 1' rot (ii) 1/3. (iii) 2/ 3. (b) (3 pts) During rolling the direction of static friction is (1)W\ (ii) down the incline. (iii) not enough info to tell. (c) (2 pts) If vcom denotes the center of mass speed, the speed of the top of the shell (point T directly across from the contact point) is equal to ( i ) UCOm ) ( ii ) vcom / 2 , r)“ "first"; (iv) not enough info to tell, (d) (2 pts) If the spherical shell and a hoop, of the same mass and radius, roll down this incline from the same heightttheihoop would come to the bottom of the incline (i) before (ii) at the same time as M’WfTN " "-KNN. ((111) may) the spherical shell. Problem 1 (24 points) A spring is initially compressed by 5.0 cm and then released with block m1 = 0.20 kg in front but not attached to it. Block m1 moves along the horizontal frictionless surface and, shortly after leaving the spring, hits block m2 = 0.30 kg. The two blocks stick together and move until stopping on the horizontal level that is h above the lower surface. The coefficient of kinetic friction of the rough surface is we = 0.30 and the distance they move along the rough surface is D 2 0.80m. (a) (4 pts) Calculate the spring constant if the speed of m1 at the relaxed spring position is 6.5 m/s. 3&3?” Come... E COW F—D—4 ................. (b) (6 pts) Calculate the speed of ml—mz after the collision. ”Fol“ Imelook wu‘lb“ Mm}: = “eh-Hm) ‘/ 35V :AMJfi Qunmp) l to 6\ a; (c) (10 pts) Calculate the height h of the higher level. Neglect the sizes of the blocks relative to the distance D. AEMeCfi : Next =3 [Gui/V4)8 9‘3 ”[3160be V7.1 : “/L‘DN : 7U M‘Vf‘vci , ‘2.“ “D :D K: Jib 3 fl 8 L7! fisg/(ofrfvwi (d) (4 pts) The center of mass of the ml-mz system while on the lower surface (Circle the right answer.) \ moving to the left at rest moving to the right. Question 2 (10 points) The figure shows a stationary rod of mass m that is held against a wall by a rope and friction between rod and wall. The uniform rod has length L and the angle between the rod and rope is 0. Circle the correct answer to the questions below. (a) (4 pts) Assume that the rotational axis is perpendicular to the rod and IS through point A Is the magnitude of the torque due to the pull on the rod by the rope (i) greater than (ii) less than the magnitue of the torque due to the rod’s weight? (b) (3 pts) The direction of the horizontal component of the force by the wall on the rod at A is (i) toward right. (ii) toward left. (iii) not enough information to decide. (0) (3 pts) If the rope is suddenly cut, the direction of the torque of gravitational force relative to the point A is , into the page. (ii) out of the page (iii) impossible to determine. Problem 2 (24 points) An object has length L, mass M, and its rotational inertia through the center of mass and axis orthogonal to the page is Icom. The object is suspended from one end as indicated in the figure. The the object is pulled to the side so that its center of mass is H = L / 5 higher and then allowed to swing as a pendulum about point A. The distance between the center of mass of the object and point A is h = L / 2. At its lowest point the object hits the ball of putty of mass m which becomes attached to it. (a) (9 pts) Calculate the magnitude and direction of the object’s angular velocity just before it hits the putty. ,— . .2 QQ‘ CN- 3‘. t“ ““30”“! Ledwe / J 3 Inca-r M 55’)?— ‘ CC~4" -21 raft): p F)? J ,IwM—IMZLr—t (b) (6 pts) What is the rotational inertia of the object—putty system about point A? ,I“+WLL N ./1~g+ ,‘J; a. _..p (c) (9 pts) Calculate the magnitude and direction of the angular velocity of the object—putty system just after the collision takes place. ,V *‘ T to LWT f3 CDMCAYNJ . I wag-’Ort : / We... Question 3 (10 points) Below are the equations for three waves traveling on separate strings. Wave 1: y(x,t) = (2.0mm) sin[(2.0m'1)a: —— (2.0 s‘l)t] Wave 2: y(:z:,t) = (3.0mm) sin[(8.0m—1):z: — (4.08—1)t] Wave 3: y(:L',t) = (1.0mm) sin[(4.0m_1)x — (8.0 s‘1)t] (a) (2.5 pts) Which wave displaces the string the most? (1) Wave 1. (iii) Wave 3. (b) (2.5 pts) Which wave has the maximum wave speed? (1) Wave 1. (ii) Wave 2. iii) Wave 3. (c) (2.5 pts) Which wave has the shortest wavelength? (1) Wave 1. (iii) Wave 3. (d) (2.5 pts) Which wave has the longest period? (ii) Wave 2. (iii) Wave 3. Problem 3 (24 points) Water is flowing with speed of 10.0 m/s through a pipe of diameter 0.2m. The water gradually descends 8.0 In as the diameter of the pipe increases to 0.4 Ill. The insert of the pipe is shown below. (a) (8 pts) What is the speed of water at the lower level? Cab-chpmgrb 5%..“ D1=O.2m (b) (16 pts) If the pressure at the upper level is 1.2 x 105 Pa, what is the pressure at the lower level? (earner-M we. 0 (‘9‘ (”9331* A2: ”hull — 32’”? L +%’ fut} “El: $4336“ 4—:2 g0} ,. :2,me ? :Q‘z‘msfi) +<£°®fi338f 8w +%‘°°°§){:to% ,Gs?)l m Question 4 (10 points) A planet travels in an elliptical orbit about a star X as shown. Q P ‘ .-o-. _.o' t. W0 .X (a) (3 pts) The magnitude of the acceleration of the planet is (i) greatest at point U. (ii) greatest at point S. iii) greatest at pinint . _ (iv) the same at all points. (b) (4 pts) At What pair of points is the speed of the planet the same? (i) P and R. (ii) W and S. (c) (3 pts) At what point is the speed of the planet minimum? (1) Point Q. (iii) Point W. Question 5 (10 points) The work that a Carnot engine delivers is one tenth of the energy it absorbs to be able to operate. (a) (2.5 pts) What is the efficiency of this engine? 10 1 1/ 10 impossible to determine (b) (2.5 pts) What is the fraction of the heat that goes to “waste” to the cold reservoir? 9 1/9 impossible to determine (c) (2.5 pts) If this engine operates such that its cold reservoir is at 200K, the temperature of the hot reservoir is roughly, 422 K 322 K 222 K impossible to determine (d) (2.5 pts) The magnitude of total change of entropy during the isothermal expansion phase is ,@ larger smaller impossible to determine than the magnitude of total change of entropy during the isothermal compression phase. Problem 4 (24 points) One mole of an ideal diatomic gas goes through the thermodynamic cycle shown in the p — V diagram below. Assume V1 2 0.1m3, V2 2 0.3 m3, and p1 = latm. (a) (6 pts) Calculate pressures p2 and p3. Aland: {Jam 7’, V, = F; V2. :7 '92.? RM ; ©"53cullwx. ?V=“e‘T—? T: 3‘4 - izt'fi’K V1 V2 2 V3 (0) (6 pts) Calculate the amount of heat absorbed (Q A) and released (Q R) during one cycle QM war may Mal/V3: (“Mafia 3( ‘5; A (2mg 22“ (9.. 3%: .- m1"? '5 am? _ {Ana 80.5 : bin : '1 CV ’51— ;(lw\{% 83' Ehx\€43éy) = "§0>5 7 ("TS : 2?: ea: 9 —~ .\ _, 3K \Qh =9 eat/- an: “’0 5M ) )(6 pts) Find the efficiency of this cycle l~—P__L\ < “$673 V5 m _____..-..........._........_....__ _ . .. . . Problem 5 (24 points) (a) (10 pts) What mass of water at 99° C must be mixed with 150 g of ice at 0° C to produce liquid water at 25" C. The heat of fusion of water is 333 kJ/kg and its specific heat is 4190 J / (kg K). Assume that the system is thermally isolated and closed. {1'4 oC (would Li a“ M. c (23’6— 342) zlq"(-‘> 3-72 k meaa‘. Mata) 7,5”C 2; 2985‘ ‘ ICE: Frames? + meneoG. 93¢; = 1W, L + I'M! (24(25’5—2233 : éS éé 3‘ —O GMch—‘i’ ©4327 fl Qw : “Q30: :0 Nu: - " 65‘66i75'3;2) Cow/#1259 ( (b) (10 pts) What is the change of entropy of the mass initially as ice when taken from ice at 0° C to water at 25° C? A3 = AS—mzuw A ASMAR—Mof 7: mil” + NI Cw ‘ (i) ‘T' T: Agra/n. 5 2’ "’38 E k (c) (4 pts) What happens to the entropy of this closed system during the process described above? Circle the right answer. @ Decreases Stays the same Impossible to determine Question 6 (10 points) Consider one mole of an ideal gas (a) (25 pts) The temperature ———————— during an adiabatic compression. CREAS \ DECREASES REMAINS THE SAME (b) 7 (2.5 pts) The temperature ———————— during a decrease in pressure at constant volume. INCREASES D REASES REMAINS THE SAME (c) (2.5 pts) The internal energy ———————— during an isothermal compression. INCREASES DECREASES REMAINS THE SAME (d) (2.5 pts) The entropy ———————— during an isothermal expansion. INCREASES DECREASES REMAINS THE SAME ...
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