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# Midterm 1 Answers - 1 First let’s analyze this system...

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Unformatted text preview: 1. First, let’s analyze this system using Newton’s law, step by step. (5 points each) i a) Write down Newton’s law and Hooke’s law. Then derive the second-order differential equation (in time) for the position x. b) Which one do you think is the correct solution of x? (Assume A is a positive value.) Explain brieﬂy why you think so. 1) x = A sin( wt) 2) x = -A sin(wt) 3) x = A cos(wt) 4) x = -A 00501)!) c) Derive the velocity v by differentiating the above expression of x by t. d) Derive the acceleration a by differentiating the above expression of v by t. e) Show that your answer in b) satisﬁes the second-order differential equation (given in a) above), if the angular frequency (0 satisfies co = i . m f) Express the amplitude A by the initial velocity v0 and a). (Recall at t= 0, v = -Va.) 65 NQ/Wl_ov;5‘, P: WK: “elk “409"” HOOVER ’. [:1ka 2. Now we are ready to express the complete solution graphically. Plot the following as a function of time. (5 points each) a) Graph of x as a ﬁmction oft. (Clearly mark the amplitude A and the period T.) b) Graph of velocity v as a function oft. (Clearly mark the initial velocity v0 and the period T.) c) Graph of acceleration a as a function of t. (Clearly mark the maximum acceleration and period T-) . d) What is the general relation between period T, frequency f and angular frequency a)? x X?- - Anemia 3. In general, it is rare that we can actually solve the second-order differential equation (i.e. 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. (5 points * 3 + 2 points * 5) a) Just after the mass is hit by a hummer at t = 0, what is the potential energy U and the kinetic energy K? So what is the total energy E? (Express it in terms of Va.) b) At x = -A, the mass st0ps. What is the potential energy U and the kinetic energy K? Then what is the total energy E? (Express it in terms of A.) c) According to the energy conservation law, the total energy given by a) and that given by b) above must be equal. Using this fact, express the amplitude A of by the initial velocity va and a). [You will find out that it is the same answer as 1-f).] d) If the initial velocity Va is doubled, i) How does the total energy change? ii) How does the amplitude change? iii) How does the period change? iv) How does the frequency change? v) How does the acceleration (just after! = 0) change? at) 5:14:21 1 my; <va '2' ‘9) F ((+15 :Jihﬂlif :éagAl (Ix/:0) 0 how“ awhéw N= EV} =2> vat-'12. k: to 4. Let’s assume the two-spring system below. The left spring has a spring constant k, where as the right spring has a spring constant 4k. Both springs are not physically glued to the mass, m. There is no space between the left spring and mass, but there is an empty space of length A between the right spring and mass (as shown in the ﬁgure). At t = 0. the mass was pushed to the left at the initial position x = -A, then it is released smoothly. As you know, it will start to oscillate like x = -A cos(wt) until the mass comes to x = 0. a) How long does it take for the mass to come to x = 0? (Express it by w.) b) After the mass comes back to x = 0, how long does it take for the mass to go from x = 0 to x = +A? (Express it by cu.) c) What is the velocity of the mass at x = +A‘? (Express it by w and A.) (1) At what location of x > 0, does the mass stop? (Express it by A.) e) Plot x as a function of t (at least for a couple of cycles of oscillation.) The shape of your graph should present the solutions of d). Please mark the exact times (in terms of co), whenx = 0 for the ﬁrst time and the second time. Spring Const.= k Spring Const.= #1 V) k C) was is ...
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