Phyic 10411 Exam III November 20, 2009 INSTRUCTIONS: Write your NAME on the front of the blue exam booklet. The exam i cloed book, and you may have only pen/pencil and a calculator (no tored equation or program and no graphing). Show all of your work in the blue book. For problem II-V, an anwer alone i worth very little credit, even if it i correct o how how you get it. Do not aume ANYTHING i obviou if it not clear, pleae ak. It hard to write thee quetion, and we often forget thing. Suggetion: Draw a diagram when poible, circle or box your final anwer, and cro out part which you do not want u to conider. Phyic 10411 Exam 3 Formula g = 9.8 m/ 2 Vector : B = B 2 x + B2 y + B2 z î = ˆx; ĵ = ŷ; ˆk = ẑ va,c = v A,B + v B,C Kinematic : Contant a x : x = x 0 + v 0,x t + 1 2 a xt 2 v x = v 0,x + a x t v 2 x = v2 0,x + 2a x(x x 0 ) 3D : v = d r dt = v xî + v yĵ + v zˆk = dx dt î + dy dt ĵ + dz dt ˆk at 2 + bt + c = 0 t = b ± b 2 4ac ; Newton : 2a a = d v dt = a xî +... = dv x dt î +... = d2 x dt 2 î +... F = Ftot = m a Work : W on object = Fon object d r U = W K = W U gravity = mg h Power = F v = K/t = W/ t = dw/dt F x, pring = kx U pring = 1 2 kx2 F centrip = mv2 r Conervation of Energy : K f + U f = K i + U i + W into ytem K lin = 1 2 mv2 F kinetic friction = µ k F normal F tatic friction µ F normal p = m v J = p = pf p i Sytem : M tot x c.m. = i m i x i p tot = M tot v c.m. v c.m. = d r c.m. dt U grav = M tot gy c.m. Rotation : α = cont. ω = ω 0 + αt ω 2 = ω 2 0 + 2α(θ θ 0) θ = θ 0 + ω 0 t + 1 2 αt2 1 rev. = 2π rad v t = rω a t = rα a r = rω 2 F net = d P c.m. dt Elatic : v i + v f = V i + V f f = ω/2π f = 1/T Page 1 of 5
Phyic 10411 Exam III November 20, 2009 I. Multiple Choice Quetion (4 point each) Pleae write the letter correponding to your anwer for each quetion in the grid tamped on the firt inide page of your blue book. No partial credit i given for thee quetion. 1. A toy truck of ma 3.45 kg i dragged up the lope AB by a motor. The lope i 36m long, and the height BC i 3.0m. There i no friction, and the acceleration i contant. The initial peed at point A i 3.5m/, and the peed at B i 5.5 m/. The average power developed by the motor i B A C (a) 17W (b) 3.9W (c) 13W (d) 130W (e) 43W 2. Two car with frictionle axle roll along identical track in uch a way that there i no energy lot to friction. Car A ha twice the ma of car B. The car follow different path; however, the finih line in either cae i exactly a height H below the tart. Which of the car finihe firt? (a) Car A (b) Car B (c) They finih at the ame time. (d) Impoible to determine (e) cantaloupe (A) (B) H H 3. Which of the following force i NOT conervative? (Hint: ketch each of the force in the x-y plane. What i W A A, the work done by the force moving around a cloed curve, if the force i conervative?) (a) F = x î + y ĵ (b) F = 5x î (c) F = k/x 2 î (d) F = ky 3 î (e) F = 2î 5 ĵ 4. A tring of force contant k = 100 N/m contract from 0.8m to 0.4m from it untretched length, pulling a ma of 5 kg. How much work i done by the pring? (a) 12 J (b) 8 J (c) 32 J (d) 24 J (e) 2 J Page 2 of 5 (Multiple Choice quetion continue )
Phyic 10411 Exam III November 20, 2009 5. A care package i dropped from a cargo plane at high altitude, and it parachute fail. Which of the following graph bet repreent the kinetic energy of the package a a function of the ditance fallen, if air reitance i taken into account? (a) (b) (c) (d) (e) Problem (20 point each) II. A 2.0 kg block i attached to a male vertical pring of pring contant k = 50 N/m. The block i lifted vertically uch that the pring i tretched 1.4 m from it untretched/uncompreed length. It i then releaed from ret. (Drawing i not to cale. Ignore the ize of the block.) a) Chooing appropriate coordinate, write down the potential energy of the ytem a a function of the height y of the block. Be ure to indicate the zero point() of your potential energie. Don t forget the pring! b) In your choen coordinate ytem, what i the total mechanical energy of the ytem? c) In your blue book, graph your potential energy function a a function of the height of the block, indicating the level of the total energy. Copy the axe provided below. Where i the minimum of the potential? d) What i the peed of the block when it i at the location where the potential energy function i at it minimum? E 1.4 m y Untretched length Page 3 of 5
Phyic 10411 Exam III November 20, 2009 III. A bullet of ma m = 5 gram travelling at 1000 m/ trike a block of ma M = 5 kg which i attached to a male pring that i initially uncompreed. The pring contant of the pring i k = 500 N/m. The bullet pae intantaneouly through the block and emerge on the other ide with a velocity of 250 m/. a) Find the recoil velocity V of the block at the moment immediately after impact, before the pring begin to compre. b) How much energy i lot in the colliion? 1000 m/ c) Auming the urface on which the block ret i k = 500 N/m M frictionle, find the ditance of maximum m compreion of the pring after the colliion. d) Auming the urface on which the block ret ha a coefficient of kinetic friction µ k = 0.2, find the x = 0 ditance of maximum compreion of the pring after the colliion. (Hint: ue the Work-Energy Theorem.) IV. A kydiver can be modeled a a cylinder of cro-ectional area A falling with intantaneou velocity v. We can ue thi a a model for the effect of air reitance on the kydiver. In order for the kydiver to move through the air, he mut puh the column of air in front of her. A he move for a time t, the column of air mut go from zero velocity up to her current velocity. Aume the air ha denity ρ, which ha unit kg/m 3. Aume that the kydiver ma i M. a) How much ma i in the column of air of length v t? b) At the end of the time interval t, the air i moving with velocity v. How much power i required to move the air? (Hint: examine your definition of power. You hould get an anwer proportional to v 3.) c) What i the drag force at thi velocity? d) Find the terminal velocity of the kydiver. e) At thi terminal velocity, how much power i gravity putting into the ytem? f) BONUS: Uing the force you found in part (c), write down but do not olve a differential equation that would give the velocity of the kydiver a a function of time if he fall from ret under the influence of gravity and aerodynamic drag. A v v t Page 4 of 5 (Problem continue )
Phyic 10411 Exam III November 20, 2009 V. Jane, whoe ma i M, need to wing acro a river of width D filled with man-eating crocodile in order to ave Tarzan from danger. However, he mut wing into a wind exerting a contant horizontal force F. The vine of length L on which he will wing make an angle θ with the vertical initially, and will make an angle φ with the vertical when he get to Tarzan (See figure). We need to tell Jane what her minimum initial peed will be in order to afely reach Tarzan. We will approach thi problem uing energy conideration. a) Since the wind force i contant, we can define a potential energy aociated with it. Why? Explain in a couple of entence. A mall ketch may help. b) Write the potential energy due to the wind force, chooing an appropriate et of coordinate. Indicate explicitly the origin of your coordinate ytem. (Hint: think gravity ideway.) c) Auming Jane leave the bank with a peed v tangent to the arc, write down her initial mechanical energy (kinetic and potential). Be ure to indicate the zero point of the gravitational potential energy. d) What are the component of Jane mechanical energy (kinetic and potential) if he jut reache Tarzan? e) Find the minimum initial peed that Jane mut have in order to arrive afely. Page 5 of 5