Energy and the Simple Harmonic Oscillator

Energy and the Simple Harmonic Oscillator tài liệu, giáo án, bài giảng , luận văn, luận án, đồ án, bài tập lớn về tất cả...

The present perfect and the simple past PEG 175-7, 182-9 (a) Fill the spaces by repeating the auxiliary used in the question, putting it into the negative where necessary. (b) Put the verb in brackets into the present perfect or the simple past tense. Have you seen that play? (a) Yes, I . . . Yes, I have. (b) Yes, I (be) there last night. Yes, I was there last night. 1 Have you wound the clock? (a) Yes, I . . . (b) Yes, I (wind) it on Monda 2 Have you ever eaten snails? (a) No, I . . . (b) Yes, I (eat) some at Tom's party last week. 3 Has she fed the dog? (a) Yes, I think she . . . (b) Yes, she (feed) him before lunch. 4 Have they repaired the road? (a) No, they . . . (b) They only (repair) part of it so far. 5 Have they done their homework? (a) Yes, they (do) it all. (b) Yes, they (do) it before they left school. 6 Have you found the matches? (a) No, I . . . (b) No, I (not find) them yet. 7 Have you made the coffee? (a) Yes, I . (b) I (make) some yesterday: we can use that. 8 Have you been here before? (a) No, I . (b) Yes, I (be) here several times. 9 Have you seen him lately? (a) No, I . . . (b) No, I (not see) him since Christmas. 10 Have you been to the opera this (a) Yes, I . . . week? (b) Yes, I (go) to Faust on Friday. For more material and information, please visit Tai Lieu Du Hoc at www.tailieuduhoc.org 11 Have you ever driven this car? (a) Yes, I (drive) it once or twice. (b) Yes, I (drive) it when you were away. 12 Has he missed his train? (a) No, he (b) Yes, he . . . It (go) five minutes ago. 13 Have they been through Customs? (a) Yes, they . . . (b) Yes, their luggage (be) examined at Dover. 14 Has he spoken to her? (a) Yes, he . . . (b) Yes, he (speak) to her on Friday. 15 Have you spent all your money? (a) No, I only (spend) half of it. (b) Yes, 1 . 16 How much have you saved (a) I (not save) anything. since Christmas? (b) I (save) Ј 3. 17 Has his temperature gone down? (a) No, it . . . (b) Yes, it (go) down last night. 18 Have you seen his garden? (a) No, I (not see) it yet. (b) I (see) the house on Monday but I (not see) the garden. 19 Have you paid the bill? (a) Yes, I . (b) Yes, I (pay) it while you were away. 20 Have you ever flown a plane? (a) No, I . . . (b) Yes, I (fly) when I was at university. 21 Has your dog ever bitten anyone? (a) Yes, he (bite) a policeman last week. (b) Yes, he (bite) me twice. 22 Have you planted your peas? (a) Yes, I (plant) them on Tuesday. (b) No, 1 . yet. 23 Has he written to the paper? (a) Yes, he . . . (b) Yes, he (write) at once. 24 Have you ever drunk vodka? (a) No, 1 . (b) I (drink) it once in Russia but I (not drink) it since. The present perfect and the simple past PEG 175-7,182-9 Put the verbs in brackets into the present perfect or the simple past tense. In some sentences the present perfect continuous (PEG 190) is also possible. 1 This is my house. ~ How long you (live) here? ~ I (live) here since 1970. 2 He (live) in London for two years and then (go) to Edinburgh. For more material and information, please visit Tai Lieu Du Hoc at www.tailieuduhoc.org 3 You (wear) your hair long when you were at school? ~ Yes, my mother (insist) on it. 4 But when I (leave) school I (cut) my hair and (wear) it short ever since. 5 Shakespeare (write) a lot of plays. 6 My brother (write) several plays. He just (finish) his second tragedy. 7 I (fly) over Loch Ness last week. ~ You (see) the Loch Ness monster? 8 I (not see) him for three years. I wonder where he is. 9 He (not smoke) for two weeks. He is trying to give it up. 10 Chopin (compose) some of his music in Majorca. 11 When he (arrive)? ~ He (arrive) at Energy and the Simple Harmonic Oscillator Energy and the Simple Harmonic Oscillator Bởi: OpenStaxCollege To study the energy of a simple harmonic oscillator, we first consider all the forms of energy it can have We know from Hooke’s Law: Stress and Strain Revisited that the energy stored in the deformation of a simple harmonic oscillator is a form of potential energy given by: PEel = kx2 Because a simple harmonic oscillator has no dissipative forces, the other important form of energy is kinetic energy KE Conservation of energy for these two forms is: KE+PEel = constant or 2 mv + kx2 = constant This statement of conservation of energy is valid for all simple harmonic oscillators, including ones where the gravitational force plays a role Namely, for a simple pendulum we replace the velocity with v = Lω, the spring constant with k = mg / L, and the displacement term with x = Lθ Thus 2 mL ω + mgLθ2 = constant In the case of undamped simple harmonic motion, the energy oscillates back and forth between kinetic and potential, going completely from one to the other as the system oscillates So for the simple example of an object on a frictionless surface attached to a spring, as shown again in [link], the motion starts with all of the energy stored in the spring As the object starts to move, the elastic potential energy is converted to kinetic energy, becoming entirely kinetic energy at the equilibrium position It is then converted 1/6 Energy and the Simple Harmonic Oscillator back into elastic potential energy by the spring, the velocity becomes zero when the kinetic energy is completely converted, and so on This concept provides extra insight here and in later applications of simple harmonic motion, such as alternating current circuits The transformation of energy in simple harmonic motion is illustrated for an object attached to a spring on a frictionless surface The conservation of energy principle can be used to derive an expression for velocity v If we start our simple harmonic motion with zero velocity and maximum displacement (x = X), then the total energy is 2 kX This total energy is constant and is shifted back and forth between kinetic energy and potential energy, at most times being shared by each The conservation of energy for this system in equation form is thus: 2 mv 1 + kx2 = kX2 Solving this equation for v yields: v=± √ mk (X2 − x2) Manipulating this expression algebraically gives: 2/6 Energy and the Simple Harmonic Oscillator v=± √ k mX √ 1− x2 X2 and so √ v = ±vmax − x2 X2 , where vmax = √ mk X From this expression, we see that the velocity is a maximum (vmax) at x = 0, as stated 2πt earlier in v(t) = − vmax sin T Notice that the maximum velocity depends on three factors Maximum velocity is directly proportional to amplitude As you might guess, the greater the maximum displacement the greater the maximum velocity Maximum velocity is also greater for stiffer systems, because they exert greater force for the same displacement This observation is seen in the expression for vmax; it is proportional to the square root of the force constant k Finally, the maximum velocity is smaller for objects that have larger masses, because the maximum velocity is inversely proportional to the square root of m For a given force, objects that have large masses accelerate more slowly A similar calculation for the simple pendulum produces a similar result, namely: √g ωmax = L θmax Determine the Maximum Speed of an Oscillating System: A Bumpy Road Suppose that a car is 900 kg and has a suspension system that has a force constant k = 6.53 × 104 N/m The car hits a bump and bounces with an amplitude of 0.100 m What is its maximum vertical velocity if you assume no damping occurs? Strategy √k We can use the expression for vmax given in vmax = m X to determine the maximum vertical velocity The variables m and k are given in the problem statement, and the maximum displacement X is 0.100 m Solution Identify known Substitute known values into vmax = √ mk X: 3/6 Energy and the Simple Harmonic Oscillator √ 6.53 × 104 N/m vmax = (0.100 m) 900 kg Calculate to find vmax= 0.852 m/s Discussion This answer seems reasonable for a bouncing car There are other ways to use conservation of energy to find vmax We could use it directly, as was done in the example featured in Hooke’s Law: Stress and Strain Revisited The small vertical displacement y of an oscillating simple pendulum, starting from its equilibrium position, is given as y(t) = a sin ωt, where a is the amplitude, ω is the angular velocity and t is the time taken Substituting 2π ω = T , we have yt = a sin ( 2πtT ) Thus, the displacement of pendulum is a function of time as shown above Also the velocity of the pendulum is given by v(t) = 2aπ T cos ( 2πtT ), so the motion of the pendulum is a function of time Check Your Understanding Why does it hurt more if your hand is snapped with a ruler than with a loose spring, even if the displacement of each system is equal? The ruler is ...Copyright © National Academy of Sciences. All rights reserved. Implementing the New Biology: Decadal Challenges Linking Food, Energy, and the Environment: Summary of a Workshop, June 3-4, 2010 http://www.nap.edu/catalog/13018.html Paula Tarnapol Whitacre, Adam P. Fagen, Jo L. Husbands, and Frances E. Sharples Planning Committee on Achieving Research Synergies for Food/Energy/ Environment Challenges: A Workshop to Explore the Potential of the “New Biology” Board on Life Sciences Division on Earth and Life Studies IMPLEMENTING THE NEW BIOLOGY Decadal Challenges Linking Food, Energy, and the Environment SU M MARY OF A WORKS HOP JUN E 3- 4 , 2010 Copyright © National Academy of Sciences. All rights reserved. Implementing the New Biology: Decadal Challenges Linking Food, Energy, and the Environment: Summary of a Workshop, June 3-4, 2010 http://www.nap.edu/catalog/13018.html THE NATIONAL ACADEMIES PRESS 500 Fifth Street, N.W. 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It is autonomous in its administration and in the selection of its mem- bers, sharing with the National Academy of Sciences the responsibility for advis- ing [...]... instead pasted together images of wind turbines, solar cells, biofuels, and electric cars When they couldn’t find clippings, they asked to sketch Dams, tidal and wave-power systems, even animal power They eagerly cobbled together fantastic totems to a gleaming future of power production As a society, we have done the same The seductive tales of wind turbines, solar cells, and biofuels foster the impression... interrogate the very idea of being for or against energy technologies at all Many energy debates arise from special interests as they posture to stake flags on the future flags adorned with the emblems of their favorite pet projects These iridescent displays have become spectacles in their own right And oh, how we do delight in a spectacle with our morning coffee Needless to say, these spectacles influence the. .. statistic, shiny teeth and all, into the limelight of government studies, textbooks, official reports, environmental statements, and into the psyches of millions of people It has become an especially powerful rhetorical device despite its misleading flaw While it’s certainly accurate to state that the quantity of solar energy hitting that small part of the desert is equivalent to the amount of energy we consume,... and others have eagerly ushered a fantasmatic array of solar devices into the spotlight, reported on their spectacular journeys into space, featured their dedicated entrepreneurs and inventors, celebrated their triumphs over dirty fossil fuels, and dared to envisage a glorious solar future for humanity The sheer magnitude of literature on the subject overwhelms— not just in newspapers, magazines, and. .. alternative to candles, disposable batteries, and kerosene lanterns, which are expensive, dirty, unreliable, and dangerous Given the appropriate socioeconomic context, solar energy can help villages raise their standards of living Radios enable   Seductive Futures farmers to monitor the weather and connect families with news and cultural events Youth who grow up with evening lighting, and thus a better... conditions from which our energy crises arise.3 As we shall discover in the chapxvi  Introduction ters to follow, these fancy energy technologies are not without side effects and limitations of their own When I speak on energy, the most frequent questions I receive are variants of “What energy technology is best?”—as if there is a straightforward answer Every energy technology causes aches and pains; shifting... special interests, and make aggressive investments in clean and renewable energy, like Google’s done with solar here in Mountain View, then we can end our addiction to oil, create millions of jobs and save the planet in the bargain –Barack Obama textbooks, Photovoltaic power generation is reliable, involves no moving parts, and the operation and maintenance costs are very low Fast set-up of doxycycline-inducible protein expression in human cell lines with a single plasmid based on Epstein– Barr virus replication and the simple tetracycline repressor Markus Bach 1 , Silke Grigat 1 , Barbara Pawlik 1 , Christian Fork 1 , Olaf Utermo ¨ hlen 2 , Sonia Pal 1 , David Banczyk 1 , Andreas Lazar 1 , Edgar Scho ¨ mig 1,3 and Dirk Gru ¨ ndemann 1,3 1 Department of Pharmacology, University of Cologne, Germany 2 Institute for Medical Microbiology, Immunology, and Hygiene, University of Cologne, Germany 3 Center for Molecular Medicine, University of Cologne (CMMC), Germany The function of human proteins is commonly analyzed by heterologous expression in cultured cell lines. Regu- lated expression, i.e. a system to switch on expression on demand, has clear advantages over constitutive expression. With constitutive expression, cells may die during antibiotic selection because of toxic effects of the expressed protein [1]. Also, for a close match of backgrounds, it is better to compare two states of a single cell line rather than two separately transfected and selected cell lines. Several widely used systems for regulated expression in mammalian cell lines are based on the tetracycline Keywords doxycycline; Epstein–Barr virus; polyadenylation; regulated protein expression; tetracycline repressor Correspondence D. Gru ¨ ndemann, Department of Pharmacology, University of Cologne, Gleueler Straße 24, 50931 Cologne, Germany Fax: +49 221 478 5022 Tel: +49 221 478 7455 E-mail: dirk.gruendemann@uni-koeln.de (Received 17 October 2006, revised 5 December 2006, accepted 5 December 2006) doi:10.1111/j.1742-4658.2006.05623.x We have developed a novel plasmid vector, pEBTetD, for full establish- ment of doxycycline-inducible protein expression by just a single transfec- tion. pEBTetD contains an Epstein–Barr virus origin of replication for stable and efficient episomal propagation in human cell lines, a cassette for continuous expression of the simple tetracycline repressor, and a cytomega- lovirus-type 2 tetracycline operator (tetO2)-tetO2 promoter. As there is no integration of vector into the genome, clonal isolation of transfected cells is not necessary. Cells are thus ready for use 1 week after transfection; this contrasts with 3–12 weeks for other systems. Adequate regulation of pro- tein expression was accomplished by abrogation of mRNA polyadenyla- tion. In northern analysis of seven cDNAs coding for transport proteins, pools of transfected human embryonic kidney 293 cells showed on ⁄ off mRNA ratios in the order of 100 : 1. Cell pools were also analyzed for regulation of protein function. With two transport proteins of the plasma membrane, the on ⁄ off activity ratios were 24 : 1 and 34 : 1, respectively. With enhanced green fluorescent protein, a 23 : 1 ratio was observed based on fluorescence intensity data from flow cytometry. The unique advantage of our system rests on the unmodified tetracycline repressor, which is less likely, by relocation upon binding of doxycycline, to cause cellular distur- bances than chimera of tetracycline repressor and eukaryotic transactiva- tion domains. Thus, in a comprehensive comparison of on- and off-states, a steady cellular background is ... oscillating 4/6 Energy and the Simple Harmonic Oscillator Section Summary • Energy in the simple harmonic oscillator is shared between elastic potential energy and kinetic energy, with the total being.. .Energy and the Simple Harmonic Oscillator back into elastic potential energy by the spring, the velocity becomes zero when the kinetic energy is completely converted, and so on This... oscillations of the building by oscillating at the same frequency as the building is being driven the driving force is transferred to the object, which 5/6 Energy and the Simple Harmonic Oscillator