mt1_solutions - Physics GB Katsushi Arisaka Last Name First...

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Unformatted text preview: Physics GB January 28, 2008 Katsushi Arisaka Last Name: First Name: Student ID No. Enrolled Lecture: 1 {llaml l' 2 {Boon} Room: HEINES 391' KNSY PV 12203 Exam Time: 11am! Noon Important Remarks: ' Sorry to tell you but during the exam, close books, close notes. no calculator is allowed; please just rely on your own brain. - Please use any open space on the exam for yotu‘ solution. Please write down how you derived concisely {winch helps you to get a partial credit even if the final answer is wrong.) " If you finish early, please be seated quietly until the exam time is over. Points: m—m -—-_ "—— “—— -—-_ “—— ——“ Question : In this exam, we explore a simple mass-spring system only. A mass (with mass m) is connected to a spring {with spring constant k) as shown below. There is no friction on the floor. At t = 0, the mass stays at the initial position x = 0, where there is no force from the spring. Then the mass is suddenly hit by a hammer (from the right to the left) and acquires the initial velocity v = we (to the left direction). Please note that 'right‘ is the positive 1: direction and ‘lefi‘ is the negative .1: direction. Hit Walter/m 1. First. let’s analyze this system using Newton‘s law, step by step. 1:) Write down Hooke’ 3 law, when the mesa is located at position x. b) By- combining it with Newton' 5 law, derive the second—order differential equation (in time) for the position x. c) Which one do you think' is the correct solution of 1:? (Assume A' [5 a positive value.) 1) .r= —A sin(e1r) 2) x— ~ —A sin(mt‘) 3) x = A costar) 4) x = —A costar) d) Derive the velocity v by differentiating the above expression of x by r. e) Derive the acceleration a by differentiating the above expression of v by r. _ i) Show that the your answer in c) satisfies the second-order differential equation {given in h) :5 above), if the angular frequency meatiafiee 1o - JE . m Express the amplitude A by the initial velocity 1)., and a1 (Recall at 1= 0, V'" — we.) :9 m 1,) Fzr'rML , flaw—+922 flrfix C) l)13“/4MCWJ 0K). ’lf’fl Hummer) 2. Now we are ready to express the complete solution graphically. Plot the following as a function of time. - :1} Graph of x as a function of r. (Clearly mark the amplitude A and the period T.) _ 1:) Graph of velocity v as a function of r. (Clearly mark the initial ‘Iurelooityr v9 and the period T.) 0} Graph of acceleration a as a function of t. (Clearly mark the period T.) d) What is the relation between angular frequency a; period T and frequency f in general? 3. In general. it is rare that we can actually solve the second-order difl'erential equation (Le. Newton‘s law). In such a case, the energy conservation law gives another powerful tool to analyze a system. So let’s try this approach next. a) What are the general forms of the potential energy U and the kinetic energy K in the case of a simple mass-spring system? (You do not have to derive it.) 1:) Just after the mass is hit by a hammer at r = 0, what is the potential energy U and the kinetic energy K"? So what is the total energy E? (Express it by v0.) c] At x = -A, the mass stops. What is the potential energy U and the kinetic energy If ? Then what is the total energy E? (Express it by A.) d} According to the energy conservation law, the total energy given by b} and that given by c) above must be equal. Using this fact, express the ampIitude A of by the initial velocity v” and or. [l’ou willfino‘ out that it is the same answer as l—g}.] a 0) oz: ml, [GEL/W2] L) M. kiwi, sew: imivfi =0 Bails/“r" 2 1__fljl__ 1 A bliu __ d W ‘ -32"; “flLW 4. Next, let’s consider the more complex system as shown below. The mass is supported by a spring on the lefi side (with Spring Constant= it). II' :1. I! 1 i in H sicall ' 4: t ifl |_L' ' ". As a result, when the mass goes to the right side (I z» o}, it will be released into curved slope {with a curvature L}. Like the previous problem, at t = (l, the mass is suddenly hit by a hammer (from the right) and acquires the initial velocity v = -va (i.e. to the lefi direction). When I < 0, the situation is exactly the same as before, where the mass has angular fi'equency in and amplitude A. A; ,5 2" ID, the mass behaves like a pendulum {with a string length L} because of the gravity mg doanards. Let' s assume that the angular frequency becomes of and amplitude heeomcsA’. (Please assume that A‘ is much shorter than L.) _ - at a) At 1: I" 0, there is a force F which brings the mass back to the left (due to gravity). Express F in terms ofm, 3,1; and x. b} At 1: 2> 0, what is the angular frequency of '? (ExpreSs it by L and g). c) What is the height of the mass (it in the picture), when it stops at x = A’ '2’ (Express it by vs and s). h i '\. «’1. t9 l. ‘. i i i t s ‘-l l i i i I Eprhgc-t- i j _ («mag ff, r-‘a 5. Let‘s assume af = 2m. Now we are ready to express the complete motion ni‘the mass graphically. a) Plot the graph (If): as a function of I. (At least plot for the time period when the mass goes left and right for a couple of times.) b) What is the relation between L, Jr, and m, if m‘ satisfies w‘ = 2m ? (Express L in terms of k, m and g}. I: 5. Let’s assume a: = Zw'. Now We are ready to express the complete motion of the mass graphically. a) PIot the graph ofx as a function of I. (At least plot for the time period when the mass goes left and right for a couple of times.) b) What is the relation betWeen L, k, and m, if m satisfies to = 200' 1’ (Express 1. in terms of k, m and s)- o) I ...
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