PHY 2060 Spring 2008 — Final ExamDO NOT TURN THE PAGE UNTIL INSTRUCTED TO DO SOInstructions: Attempt all ten questions, each of which carries a maximum of 10 points.Write your solution below each question, continuing on additional paper if necessary. Pleasetry to write neatly!You will receive credit only for knowledge and understanding that you demonstrate inyour written solutions. It is in your best interest to write down something relevant forevery question, even if you can’t provide a complete answer. To maximize your score, youshould briefly explain your reasoning and show all working. (This may benefit you evenin the case of the multiple-choice Question 1.) Give all final algebraic answers in terms ofvariables defined in the problem, g (the acceleration due to gravity near the Earth’s surface),G (the gravitational constant), and/or c (the speed of light). For numerical problems, takeg =10m/s2, G =6.7 × 10−11Nm2/kg2,andc =3.0 × 108m/s.1. For each of the two parts of this question, place a check next to one answer only.(a) A projectile is fired from ground level over level ground with an initial velocitythat has a vertical component of 20 m/s and a horizontal component of 30 m/s.Using g =10 m/s2, the distance from launching to landing points isi. 40 mii. 60 miii. 80 miv. 120 mv. 180 m(b) A certain spring elongates 9.0 mm when it is suspended vertically and a block ofmass M is hung on it. The natural angular frequency of this block-spring systemi. is 0.088 rad/s.ii. is 33 rad/s.iii. is 200 rad/s.iv. is 1140 rad/s.v. cannot be computed unless the value of M is given.2. The coordinates x and y (in meters) of a particle as a function of time t (in seconds)are x =3t2and y =16t − 4t2.(a) Find the particle’s velocity and acceleration at time t.(b) Find the particle’s smallest speed.3. The planet Mars has a mass of 6.4×1023kg and a radius of 3,400 km. The first mannedspace expedition to Mars makes the remarkable discovery that the planet is a sphericalshell only 1 km thick. Find both (i) the force in newtons on the expedition’s 16,000-kgspaceship and (ii) the binding energy in joules of the Mars-spaceship system, when thespaceship is located ...(a) 500 km from the center of Mars.(b) 5,000 km from the center of Mars.4. A thin cylindrical pipe of mass M and radius R is held by a pair of ideal ropes ina horizontal orientation against a frictionless wall, as shown in the pair of side viewsbelow. The pipe walls are of uniform thickness. Each of the ropes is of length L and isattached to the pipe in such a way that the rope points through the pipe’s cylindricalaxis. Find the tension in each rope and the magnitude of the force exerted by the wallon the pipe.RLside viewfront view5. A ladder of length L and negligible mass leans against a frictionless wall. The anglebetween the ladder and the horizontal ground is θ. The coefficient of static frictionbetween the ladder and the ground is µs. A person of mass m starts slowly up theladder. What fraction of the way up the ladder can the person reach before the ladderbecomes unstable?6. A pulley consists of a uniform disk of mass m and radius L/10 thatcan rotate freely about a horizontal, frictionless axle. A uniformrope of length L and mass M is hung over the pulley, with half therope’s length dangling straight down on one side of the pulley, andthe rest of the rope either in contact with the pulley or hangingstraight down on the other side, as shown in the diagram. Thissystem is released from rest. How fast will the bottom end of therope be traveling when it has traveled a distance L/10 (withouthitting any nearby objects)? Assume that the rope does not sliparound the perimeter of the pulley.L/2RR = L/1027. Alice measures event B to occur 2.0 µs after event A and at a displacement ∆x =1.0 km from A. Alice also observes Bob to be traveling at a constant velocity along herx axis. Bob measures events A and B to be simultaneous.(a) Is it possible for event A to have caused event B?(b) How fast must Bob be moving in Alice’s frame?(c) Suppose that Alice measures event C to occur 2.0 µs after event A. What is thesmallest distance |∆x| by which events A and C must be separated for it to bepossible for any inertial observer to measure the events to occur simultaneously?8. Tarzan, sitting on top of a large termite mound, sees Jane in mortal danger standingon the jungle floor. Tarzan swings in a quarter circle on the end of an ideal (massless,unstretchable) vine of length 15.0 m and grabs Jane, as shown in the left-hand diagram.Unfortunately, Tarzan has failed to spot a low-lying branch located 13.0 m directlybelow Tarzan’s pivot point and 2.0 m directly above where Jane was initially standing.The vine catches on the branch, causing Tarzan and Jane to rise to the position shownin the right-hand diagram. Given that Tarzan has a mass of 80 kg and Jane’s massis 50 kg, find the tension in the vine at the moment shown in the right-hand figure.(Treat Tarzan and Jane as point masses.)TJbranchpivotT+J9. A pendulum consists of a thin, uniform rod of mass M and length L,towhichasmallball of mass 2M is firmly attached. The rod is pivoted freely at one end. How far fromthe pivot should the ball be attached so that the pendulum has the shortest possibleperiod?310. A mass m sits on a horizontal surface, connected to a fixed point by a horizontal,ideal spring of constant k, as shown in the diagram. The coefficient of kinetic frictionbetween the mass and the surface is µk. The mass is released from rest at a pointwhere the spring is extension is x0> 0. The spring force is sufficient to overcomestatic friction, so the spring starts in motion.(a) Find the spring displacement x1at the firsttime the mass comes to rest.(b) Assuming that the spring force at x = x1issufficient to overcome static friction, findthe spring displacement x2at the secondtime the mass comes to