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# mt1a_solution - Physics 6B Katsushi Arisaka First...

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Unformatted text preview: Physics 6B January 26, 2009 Katsushi Arisaka First Midterm (Type A) Last Name: Z i RISA k A First Name: (6 WW Student ID N0. N4 A Enrolled Lecture: 1 (11am) / 2 (Noon) Exam Time: 11am / Noon Important Remarks: I Sorry to tell you but during the exam, close books, close notes, no calculator is allowed; please just rely on your own brain. 0 Please use any open space on the exam for your solution. Please write down how you derived concisely (which helps you to get a partial credit, even if the ﬁnal answer is wrong.) I Even if you ﬁnish early, please be seated quietly until the exam time is over. Points: -—-_ ---— 6t><3+8 t =26 - 2 1-.— 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) and acquires the initial velocity v = -v0 (to the left direction, Va is positive value). Please note that “right” is positive direction of x. A Spring Const.= k Mass 2 m , - ./ fly ..‘f. o .- A _. .- -‘ , _ 3' r / ,w J _.r a v r / v--va <— ‘ . 1. First, let’s analyze this system using Newton’s law, step by step. (5 points each) a) Write down the Hooke’s law, when the mass is located at position x. b) By combining it with Newton’s law, derive the second-order differential equation (in time) for the position x. c) Which one do you think is the correct solution of x? (Assume A is a positive value.) 1) x = A sin( (at) 2) x = -A sin((at) 3) x = A cos( wt) 4) x = —A cos(wt) d) Derive the velocity v by differentiating the above expression of x by t. e) Derive the acceleration a by differentiating the above expression of v by t. t) Show that the your answer in c) satisﬁes the second-order differential equation (given in b) above), if the angular frequency (0 satisﬁes a) = P . m g) Express the amplitude A by the initial velocity v0 and a1 (Recall at t = 0, v = -v0.) 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 function oft. (Clearly mark the amplitude A and the period T.) b) Graph of velocity v as a function of t. (Clearly mark the initial velocity v0 and the period T.) c) Graph of acceleration a as a function of 1‘. (Clearly mark the period T.) 0*) 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. (6 points each) 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 stops. 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 v0 and a). [You will ﬁnd out that it is the same answer as l-g).] d) At x = %A , what is the velocity v ? (Express it only by v0.) ‘ (8 points) 4. When the mass comes back to x = 0 for the ﬁrst time, the second spring (with the spring constant 8k) is inserted smoothly It is physically connected to the mass as shown below. The insertion is so smooth that the velocity of the mass does not change iust betore and just after the insertion of the second spring. (4 points each) a) What is the total energy of the new system E’ at x = 0 just after the second spring is inserted? (Express E’ by E.) b) What is the “effective” spring constant of the new system k ’7 (Express k’ by k.) c) What is the new angular frequency co"? (ExpresSaJ’ by on) d) What is the new period T’? (Express T’ by T.) e) What is the new amplitude A ? (Express A’ by A. ) f) Plot x as a function of t (at least for 0 < t < T. Clearly mark T and A .) The shape of your graph should present the solutions of d) and e) well Spring Const.= k Spring Const.= 8k TYFQ B WVVVV a E’-’E (1 233.37‘“ I ) /_. z )T UI'W:4 b)}c-\€‘HW€ ‘lﬂe A ...
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mt1a_solution - Physics 6B Katsushi Arisaka First...

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