<> stream WebAuthor: ANA Subject: Set #4 Created Date: 11/19/2001 3:08:22 PM 16.4 The Simple Pendulum - College Physics 2e | OpenStax Solutions endobj 21 0 obj The only things that affect the period of a simple pendulum are its length and the acceleration due to gravity. >> 666.7 666.7 666.7 666.7 611.1 611.1 444.4 444.4 444.4 444.4 500 500 388.9 388.9 277.8 WebThe simple pendulum system has a single particle with position vector r = (x,y,z). 8.1 Pendulum experiments Activity 1 Your intuitive ideas To begin your investigation you will need to set up a simple pendulum as shown in the diagram. 820.5 796.1 695.6 816.7 847.5 605.6 544.6 625.8 612.8 987.8 713.3 668.3 724.7 666.7 ollB;% !JA6Avls,/vqnpPw}o@g `FW[StFb s%EbOq#!!!h#']y\1FKW6 Why does this method really work; that is, what does adding pennies near the top of the pendulum change about the pendulum? /Type/Font /LastChar 196 Want to cite, share, or modify this book? 877 0 0 815.5 677.6 646.8 646.8 970.2 970.2 323.4 354.2 569.4 569.4 569.4 569.4 569.4 /FirstChar 33 61) Two simple pendulums A and B have equal length, but their bobs weigh 50 gf and l00 gf respectively. 285.5 799.4 485.3 485.3 799.4 770.7 727.9 742.3 785 699.4 670.8 806.5 770.7 371 528.1 WebAustin Community College District | Start Here. Consider the following example. /BaseFont/JOREEP+CMR9 750 708.3 722.2 763.9 680.6 652.8 784.7 750 361.1 513.9 777.8 625 916.7 750 777.8 /LastChar 196 351.8 935.2 578.7 578.7 935.2 896.3 850.9 870.4 915.7 818.5 786.1 941.7 896.3 442.6 Notice the anharmonic behavior at large amplitude. That means length does affect period. endobj /FirstChar 33 Solution: 624.1 928.7 753.7 1090.7 896.3 935.2 818.5 935.2 883.3 675.9 870.4 896.3 896.3 1220.4 473.8 498.5 419.8 524.7 1049.4 524.7 524.7 524.7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 endobj %PDF-1.4 The displacement ss is directly proportional to . /Name/F1 Single and Double plane pendulum 795.8 795.8 649.3 295.1 531.3 295.1 531.3 295.1 295.1 531.3 590.3 472.2 590.3 472.2 787 0 0 734.6 629.6 577.2 603.4 905.1 918.2 314.8 341.1 524.7 524.7 524.7 524.7 524.7 /FontDescriptor 32 0 R >> /FThHh!nmoF;TSooevBFN""(+7IcQX.0:Pl@Hs (@Kqd(9)\ (jX Play with one or two pendulums and discover how the period of a simple pendulum depends on the length of the string, the mass of the pendulum bob, and the amplitude of the swing. They attached a metal cube to a length of string and let it swing freely from a horizontal clamp. >> Adding pennies to the Great Clock shortens the effective length of its pendulum by about half the width of a human hair. Simplify the numerator, then divide. The length of the second pendulum is 0.4 times the length of the first pendulum, and the, second pendulum is 0.9 times the acceleration of gravity, The length of the cord of the first pendulum, The length of cord of the second pendulum, Acceleration due to the gravity of the first pendulum, Acceleration due to gravity of the second pendulum, he comparison of the frequency of the first pendulum (f. Hertz. The Island Worksheet Answers from forms of energy worksheet answers , image source: www. Resonance of sound wave problems and solutions, Simple harmonic motion problems and solutions, Electric current electric charge magnetic field magnetic force, Quantities of physics in the linear motion. Pendulums 877 0 0 815.5 677.6 646.8 646.8 970.2 970.2 323.4 354.2 569.4 569.4 569.4 569.4 569.4 /FontDescriptor 26 0 R Simple Pendulum What is the period of the Great Clock's pendulum? <> /FirstChar 33 g The period of the Great Clock's pendulum is probably 4seconds instead of the crazy decimal number we just calculated. WebFor periodic motion, frequency is the number of oscillations per unit time. The heart of the timekeeping mechanism is a 310kg, 4.4m long steel and zinc pendulum. 0 0 0 0 0 0 0 0 0 0 0 0 675.9 937.5 875 787 750 879.6 812.5 875 812.5 875 0 0 812.5 1. 791.7 777.8] Physics 1 First Semester Review Sheet, Page 2. Pennies are used to regulate the clock mechanism (pre-decimal pennies with the head of EdwardVII). 750 758.5 714.7 827.9 738.2 643.1 786.2 831.3 439.6 554.5 849.3 680.6 970.1 803.5 endobj 20 0 obj /FontDescriptor 23 0 R If, is the frequency of the first pendulum and, is the frequency of the second pendulum, then determine the relationship between, Based on the equation above, can conclude that, ased on the above formula, can conclude the length of the, (l) and the acceleration of gravity (g) impact the period of, determine the length of rope if the frequency is twice the initial frequency. (* !>~I33gf. /Widths[314.8 527.8 839.5 786.1 839.5 787 314.8 419.8 419.8 524.7 787 314.8 367.3 What is the answer supposed to be? Solution: Once a pendulum moves too fast or too slowly, some extra time is added to or subtracted from the actual time. /Type/Font 708.3 795.8 767.4 826.4 767.4 826.4 0 0 767.4 619.8 590.3 590.3 885.4 885.4 295.1 What is the period on Earth of a pendulum with a length of 2.4 m? Substitute known values into the new equation: If you are redistributing all or part of this book in a print format, Representative solution behavior and phase line for y = y y2. <>>> /Subtype/Type1 777.8 777.8 1000 1000 777.8 777.8 1000 777.8] Websimple harmonic motion. 24/7 Live Expert. /Name/F11 UNCERTAINTY: PROBLEMS & ANSWERS /Type/Font They recorded the length and the period for pendulums with ten convenient lengths. 4 0 obj 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 706.4 938.5 877 781.8 754 843.3 815.5 877 815.5 << 742.3 799.4 0 0 742.3 599.5 571 571 856.5 856.5 285.5 314 513.9 513.9 513.9 513.9 Two simple pendulums are in two different places. % 18 0 obj B ased on the above formula, can conclude the length of the rod (l) and the acceleration of gravity (g) impact the period of the simple pendulum. /BaseFont/AVTVRU+CMBX12 l(&+k:H uxu {fH@H1X("Esg/)uLsU. 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 << endobj The problem said to use the numbers given and determine g. We did that. Solution: The period and length of a pendulum are related as below \begin{align*} T&=2\pi\sqrt{\frac{\ell}{g}} \\\\3&=2\pi\sqrt{\frac{\ell}{9.8}}\\\\\frac{3}{2\pi}&=\sqrt{\frac{\ell}{9.8}} \\\\\frac{9}{4\pi^2}&=\frac{\ell}{9.8}\\\\\Rightarrow \ell&=9.8\times\left(\frac{9}{4\pi^2}\right)\\\\&=2.23\quad{\rm m}\end{align*} The frequency and periods of oscillations in a simple pendulum are related as $f=1/T$. endobj This method isn't graphical, but I'm going to display the results on a graph just to be consistent. The most popular choice for the measure of central tendency is probably the mean (gbar). /Widths[622.5 466.3 591.4 828.1 517 362.8 654.2 1000 1000 1000 1000 277.8 277.8 500 <> 687.5 312.5 581 312.5 562.5 312.5 312.5 546.9 625 500 625 513.3 343.8 562.5 625 312.5 Examples of Projectile Motion 1. 19 0 obj endobj Solution: The length $\ell$ and frequency $f$ of a simple pendulum are given and $g$ is unknown. If you need help, our customer service team is available 24/7. Use this number as the uncertainty in the period. The period of a pendulum on Earth is 1 minute. WebThe simple pendulum system has a single particle with position vector r = (x,y,z). /LastChar 196 Dowsing ChartsUse this Chart if your Yes/No answers are What would be the period of a 0.75 m long pendulum on the Moon (g = 1.62 m/s2)? There are two basic approaches to solving this problem graphically a curve fit or a linear fit. << WebThe simple pendulum is another mechanical system that moves in an oscillatory motion. The length of the cord of the first pendulum (l1) = 1, The length of cord of the second pendulum (l2) = 0.4 (l1) = 0.4 (1) = 0.4, Acceleration due to the gravity of the first pendulum (g1) = 1, Acceleration due to gravity of the second pendulum (g2) = 0.9 (1) = 0.9, Wanted: The comparison of the frequency of the first pendulum (f1) to the second pendulum (f2). In trying to determine if we have a simple harmonic oscillator, we should note that for small angles (less than about 1515), sinsin(sinsin and differ by about 1% or less at smaller angles). 306.7 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 306.7 306.7 11 0 obj 656.3 625 625 937.5 937.5 312.5 343.8 562.5 562.5 562.5 562.5 562.5 849.5 500 574.1 /Subtype/Type1 /FontDescriptor 11 0 R Students calculate the potential energy of the pendulum and predict how fast it will travel. endobj /W [0 [777.832 0 0 250 0 408.2031 500 0 0 777.832 180.1758 333.0078 333.0078 0 563.9648 250 333.0078 250 277.832] 19 28 500 29 [277.832] 30 33 563.9648 34 [443.8477 920.8984 722.168 666.9922 666.9922 722.168 610.8398 556.1523 0 722.168 333.0078 389.1602 722.168 610.8398 889.1602 722.168 722.168 556.1523 722.168 0 556.1523 610.8398 722.168 722.168 943.8477 0 0 610.8398] 62 67 333.0078 68 [443.8477 500 443.8477 500 443.8477 333.0078 500 500 277.832 277.832 500 277.832 777.832] 81 84 500 85 [333.0078 389.1602 277.832 500 500 722.168 500 500 443.8477] 94 130 479.9805 131 [399.9023] 147 [548.8281] 171 [1000] 237 238 563.9648 242 [750] 520 [582.0313] 537 [479.0039] 550 [658.2031] 652 [504.8828] 2213 [526.3672]]>> 295.1 826.4 501.7 501.7 826.4 795.8 752.1 767.4 811.1 722.6 693.1 833.5 795.8 382.6 Bonus solutions: Start with the equation for the period of a simple pendulum. %PDF-1.5 \begin{gather*} T=2\pi\sqrt{\frac{2}{9.8}}=2.85\quad {\rm s} \\ \\ f=\frac{1}{2.85\,{\rm s}}=0.35\quad {\rm Hz}\end{gather*}. 874 706.4 1027.8 843.3 877 767.9 877 829.4 631 815.5 843.3 843.3 1150.8 843.3 843.3 Then, we displace it from its equilibrium as small as possible and release it. The mass does not impact the frequency of the simple pendulum. The angular frequency formula (10) shows that the angular frequency depends on the parameter k used to indicate the stiffness of the spring and mass of the oscillation body. 5. 1 0 obj 597.2 736.1 736.1 527.8 527.8 583.3 583.3 583.3 583.3 750 750 750 750 1044.4 1044.4 277.8 305.6 500 500 500 500 500 750 444.4 500 722.2 777.8 500 902.8 1013.9 777.8 2022 Practice Exam 1 Mcq Ap Physics Answersmotorola apx Solution Austin Community College District | Start Here. Get There. All Physics C Mechanics topics are covered in detail in these PDF files. /Contents 21 0 R Mathematical %PDF-1.2 In the case of a massless cord or string and a deflection angle (relative to vertical) up to $5^\circ$, we can find a simple formula for the period and frequency of a pendulum as below \[T=2\pi\sqrt{\frac{\ell}{g}}\quad,\quad f=\frac{1}{2\pi}\sqrt{\frac{g}{\ell}}\] where $\ell$ is the length of the pendulum and $g$ is the acceleration of gravity at that place. %PDF-1.5 /Name/F8 /Name/F9 What is the acceleration due to gravity in a region where a simple pendulum having a length 75.000 cm has a period of 1.7357 s? 1444.4 555.6 1000 1444.4 472.2 472.2 527.8 527.8 527.8 527.8 666.7 666.7 1000 1000 Differential equation @ @y ss~P_4qu+a" ' 9y c&Ls34f?q3[G)> `zQGOxis4t&0tC: pO+UP=ebLYl*'zte[m04743C 3d@C8"P)Dp|Y Support your local horologist. 36 0 obj xY[~pWE4i)nQhmVcK{$9_,yH_,fH|C/8I}~\pCIlfX*V$w/;,W,yPP YT,*} 4X,8?._,zjH4Ib$+p)~%B-WqmQ-v9Z^85'))RElMaBa)L^4hWK=;fQ}|?X3Lzu5OTt2]/W*MVr}j;w2MSZTE^*\ h 62X]l&S:O-n[G&Mg?pp)$Tt%4r6fm=4e"j8 495.7 376.2 612.3 619.8 639.2 522.3 467 610.1 544.1 607.2 471.5 576.4 631.6 659.7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 663.6 885.4 826.4 736.8 460 664.4 463.9 485.6 408.9 511.1 1022.2 511.1 511.1 511.1 0 0 0 0 0 0 0 0 0 0 0 In part a ii we assumed the pendulum would be used in a working clock one designed to match the cultural definitions of a second, minute, hour, and day. 384.3 611.1 611.1 611.1 611.1 611.1 896.3 546.3 611.1 870.4 935.2 611.1 1077.8 1207.4 Solution 500 500 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 625 833.3 /Subtype/Type1 Let us define the potential energy as being zero when the pendulum is at the bottom of the swing, = 0 . /BaseFont/SNEJKL+CMBX12 /FontDescriptor 8 0 R 323.4 354.2 600.2 323.4 938.5 631 569.4 631 600.2 446.4 452.6 446.4 631 600.2 815.5 Web25 Roulette Dowsing Charts - Pendulum dowsing Roulette Charts PendulumDowsing101 $8. 10 0 obj 12 0 obj WebMISN-0-201 7 Table1.Usefulwaverelationsandvariousone-dimensional harmonicwavefunctions.Rememberthatcosinefunctions mayalsobeusedasharmonicwavefunctions. /BaseFont/EKBGWV+CMR6 Projectile motion problems and answers Problem (1): A person kicks a ball with an initial velocity of 15\, {\rm m/s} 15m/s at an angle of 37 above the horizontal (neglect the air resistance). /Type/Font <> endstream /LastChar 196 298.4 878 600.2 484.7 503.1 446.4 451.2 468.8 361.1 572.5 484.7 715.9 571.5 490.3 Physics 1 Lab Manual1Objectives: The main objective of this lab <> << 324.7 531.3 590.3 295.1 324.7 560.8 295.1 885.4 590.3 531.3 590.3 560.8 414.1 419.1 743.3 743.3 613.3 306.7 514.4 306.7 511.1 306.7 306.7 511.1 460 460 511.1 460 306.7 In this problem has been said that the pendulum clock moves too slowly so its time period is too large. stream stream A classroom full of students performed a simple pendulum experiment. 314.8 787 524.7 524.7 787 763 722.5 734.6 775 696.3 670.1 794.1 763 395.7 538.9 789.2 /Annots [<>>> <>>> <>>> <>>> <>>> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <>] This is for small angles only. /Subtype/Type1 Simple Pendulum Simple Pendulum Solution; Find the maximum and minimum values of \(f\left( {x,y} \right) = 8{x^2} - 2y\) subject to the constraint \({x^2} + {y^2} = 1\). The equation of period of the simple pendulum : T = period, g = acceleration due to gravity, l = length of cord. This leaves a net restoring force back toward the equilibrium position at =0=0. They recorded the length and the period for pendulums with ten convenient lengths. /BaseFont/CNOXNS+CMR10 << /LastChar 196 Pendulum 2 has a bob with a mass of 100 kg100 kg. Some simple nonlinear problems in mechanics, for instance, the falling of a ball in fluid, the motion of a simple pendulum, 2D nonlinear water waves and so on, are used to introduce and examine the both methods. /Subtype/Type1 if(typeof ez_ad_units != 'undefined'){ez_ad_units.push([[300,250],'physexams_com-large-mobile-banner-2','ezslot_8',133,'0','0'])};__ez_fad_position('div-gpt-ad-physexams_com-large-mobile-banner-2-0'); Problem (10): A clock works with the mechanism of a pendulum accurately. Thus, The frequency of this pendulum is \[f=\frac{1}{T}=\frac{1}{3}\,{\rm Hz}\], Problem (3): Find the length of a pendulum that has a frequency of 0.5 Hz. Period is the goal. How accurate is this measurement? Creative Commons Attribution License 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 643.8 839.5 787 710.5 682.1 763 734.6 787 734.6 593.8 500 562.5 1125 562.5 562.5 562.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 The period of a simple pendulum is described by this equation. <> Solution: The frequency of a simple pendulum is related to its length and the gravity at that place according to the following formula \[f=\frac {1}{2\pi}\sqrt{\frac{g}{\ell}}\] Solving this equation for $g$, we have \begin{align*} g&=(2\pi f)^2\ell\\&=(2\pi\times 0.601)^2(0.69)\\&=9.84\quad {\rm m/s^2}\end{align*}, Author: Ali Nemati WebMass Pendulum Dynamic System chp3 15 A simple plane pendulum of mass m 0 and length l is suspended from a cart of mass m as sketched in the figure. << <> The length of the cord of the simple pendulum (l) = 1 meter, Wanted: determine the length of rope if the frequency is twice the initial frequency. The Pendulum Brought to you by Galileo - Georgetown ISD 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 777.8 444.4 444.4 Pendulum B is a 400-g bob that is hung from a 6-m-long string. 783.4 872.8 823.4 619.8 708.3 654.8 0 0 816.7 682.4 596.2 547.3 470.1 429.5 467 533.2 >> Pendulums - Practice The Physics Hypertextbook The short way F /Subtype/Type1 351.8 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 351.8 351.8 Given that $g_M=0.37g$. Length and gravity are given. endobj << /Type /XRef /Length 85 /Filter /FlateDecode /DecodeParms << /Columns 5 /Predictor 12 >> /W [ 1 3 1 ] /Index [ 18 54 ] /Info 16 0 R /Root 20 0 R /Size 72 /Prev 140934 /ID [<8a3b51e8e1dcde48ea7c2079c7f2691d>] >> How long is the pendulum? 611.1 798.5 656.8 526.5 771.4 527.8 718.7 594.9 844.5 544.5 677.8 762 689.7 1200.9 Physics 6010, Fall 2010 Some examples. Constraints and Websome mistakes made by physics teachers who retake models texts to solve the pendulum problem, and finally, we propose the right solution for the problem fashioned as on Tipler-Mosca text (2010). /Subtype/Type1 /Name/F4 21 0 obj These NCERT Solutions provide you with the answers to the question from the textbook, important questions from previous year question papers and sample papers. << moving objects have kinetic energy. Modelling of The Simple Pendulum and It Is Numerical Solution By the end of this section, you will be able to: Pendulums are in common usage. /FontDescriptor 11 0 R Web3 Phase Systems Tutorial No 1 Solutions v1 PDF Lecture notes, lecture negligence Summary Small Business And Entrepreneurship Complete - Course Lead: Tom Coogan Advantages and disadvantages of entry modes 2 Lecture notes, lectures 1-19 - materials slides Frustration - Contract law: Notes with case law 0 0 0 0 0 0 0 0 0 0 777.8 277.8 777.8 500 777.8 500 777.8 777.8 777.8 777.8 0 0 777.8 xZYs~7Uj)?$e'VP$DJOtn/ *ew>>D/>\W/O0ttW1WtV\Uwizb va#]oD0n#a6pmzkm7hG[%S^7@[2)nG%,acV[c{z$tA%tpAi59t> @SHKJ1O(8_PfG[S2^$Y5Q }(G'TcWJn{ 0":4htmD3JaU?n,d]!u0"] oq$NmF~=s=Q3K'R1>Ve%w;_n"1uAtQjw8X?:(_6hP0Kes`@@TVy#Q$t~tOz2j$_WwOL. PHET energy forms and changes simulation worksheet to accompany simulation. endstream are not subject to the Creative Commons license and may not be reproduced without the prior and express written A classroom full of students performed a simple pendulum experiment. For angles less than about 1515, the restoring force is directly proportional to the displacement, and the simple pendulum is a simple harmonic oscillator. endobj One of the authors (M. S.) has been teaching the Introductory Physics course to freshmen since Fall 2007. endobj 527.8 314.8 524.7 314.8 314.8 524.7 472.2 472.2 524.7 472.2 314.8 472.2 524.7 314.8 /Type/Font /Subtype/Type1 A simple pendulum with a length of 2 m oscillates on the Earths surface. Problem (5): To the end of a 2-m cord, a 300-g weight is hung. Problem (6): A pendulum, whose bob has a mass of $2\,{\rm g}$, is observed to complete 50 cycles in 40 seconds. /FirstChar 33 endstream 542.4 542.4 456.8 513.9 1027.8 513.9 513.9 513.9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Our mission is to improve educational access and learning for everyone. /Subtype/Type1 /BaseFont/WLBOPZ+CMSY10 WebEnergy of the Pendulum The pendulum only has gravitational potential energy, as gravity is the only force that does any work. 306.7 766.7 511.1 511.1 766.7 743.3 703.9 715.6 755 678.3 652.8 773.6 743.3 385.6 /Type/Font 277.8 500] 481.5 675.9 643.5 870.4 643.5 643.5 546.3 611.1 1222.2 611.1 611.1 611.1 0 0 0 0 5 0 obj The worksheet has a simple fill-in-the-blanks activity that will help the child think about the concept of energy and identify the right answers. 692.5 323.4 569.4 323.4 569.4 323.4 323.4 569.4 631 507.9 631 507.9 354.2 569.4 631 i.e. Simple pendulum problems and solutions PDF WebThe essence of solving nonlinear problems and the differences and relations of linear and nonlinear problems are also simply discussed. This is why length and period are given to five digits in this example. Solution: (a) the number of complete cycles $N$ in a specific time interval $t$ is defined as the frequency $f$ of an oscillatory system or \[f=\frac{N}{t}\] Therefore, the frequency of this pendulum is calculated as \[f=\frac{50}{40\,{\rm s}}=1.25\, {\rm Hz}\] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 458.3 458.3 416.7 416.7 314.8 524.7 524.7 524.7 524.7 524.7 524.7 524.7 524.7 524.7 524.7 524.7 314.8 314.8 >> As with simple harmonic oscillators, the period TT for a pendulum is nearly independent of amplitude, especially if is less than about 1515. Calculate the period of a simple pendulum whose length is 4.4m in London where the local gravity is 9.81m/s2. >> 0 0 0 0 0 0 0 0 0 0 0 0 675.9 937.5 875 787 750 879.6 812.5 875 812.5 875 0 0 812.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 312.5 312.5 342.6 If the length of the cord is increased by four times the initial length, then determine the period of the harmonic motion. @bL7]qwxuRVa1Z/. HFl`ZBmMY7JHaX?oHYCBb6#'\ }! ))NzX2F A "seconds pendulum" has a half period of one second. 6 problem-solving basics for one-dimensional kinematics, is a simple one-dimensional type of projectile motion in . An object is suspended from one end of a cord and then perform a simple harmonic motion with a frequency of 0.5 Hertz. This method for determining pendulum /Name/F2 33 0 obj <> B]1 LX&? /Name/F3 they are also just known as dowsing charts . g /Type/Font The motion of the cart is restrained by a spring of spring constant k and a dashpot constant c; and the angle of the pendulum is restrained by a torsional spring of This PDF provides a full solution to the problem.
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