Simple Harmonic Motion

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1 Siple Haronic Motion Physics Enhanceent Prograe for Gifted Students The Hong Kong Acadey for Gifted Education and Departent of Physics, HKBU Departent of Physics

No atter what the direction of the displaceent, the force always acts in a direction to restore the syste to its

2 Siple haronic otion In echanical physics, siple haronic otion is a type of periodic otion where the restoring force is directly proportional to the displaceent. No atter what the direction of the displaceent, the force always acts in a direction to restore the syste to its equilibriu position. It can serve as a atheatical odel of a variety of otions, such as the oscillation of a spring, otion of a siple pendulu as well as olecular vibration. Departent of Physics

3 Matheatics of siple haronic otion Siple haronic otion is a type of periodic otion which can use atheatical odel to epress it. ( t) cos( t ) T f Displaceent(position) (t) Aplitude Phase Angular frequency Frequency f Period T Departent of Physics 3

4 Matheatics of siple haronic otion Since the otion returns to its initial value after one period T, cos( t ) cos[ ( t T) )] t ( t T) T f T Departent of Physics 4

5 Matheatics of siple haronic otion Velocity d d v( t) [ cos( t )] dt dt v( t) sin( t ) Velocity aplitude v Acceleration dv d a( t) [ sin( t )] dt dt a( t) cos( t ) ( t) Acceleration aplitude a Equation of otion d dt This equation of otion will be very useful in identifying siple haronic otion and its frequency. Departent of Physics 5

For one-diensional siple haronic otion, the equation of otion, which is a second-order linear ordinary differential equation with constant

6 Siple haronic otion of spring Siple haronic otion of a ass on a spring is subject to the linear elastic restoring force given by Hooe's aw. The otion is sinusoidal in tie and deonstrates a single resonant frequency. For one-diensional siple haronic otion, the equation of otion, which is a second-order linear ordinary differential equation with constant coefficients, could be obtained by eans of Newton's second law and Hooe's law. F F a d dt where is the inertial ass of the oscillating body, is its displaceent fro the equilibriu, and is the spring constant. d dt Departent of Physics 6

Siple haronic otion of spring Coparing with the equation of otion for siple haronic otion, Angular frequency Period T T Siple haronic otion is the

7 Siple haronic otion of spring Coparing with the equation of otion for siple haronic otion, Angular frequency Period T T Siple haronic otion is the otion eecuted by a particle of ass subject to a force that is proportional to the displaceent of the particle but opposite in sign. Departent of Physics 7

8 Eaples A bloc whose ass is 68 g is fastened to a spring whose spring constant is 65 N -. The bloc is pulled a distance = c fro its equilibriu position at = on a frictionless surface and released fro rest at t =. What are the angular frequency, the frequency, and the period of the resulting oscillation? What is the aplitude of the oscillation? What is the aiu speed of the oscillating bloc? What is the agnitude of the aiu acceleration of the bloc? What is the phase constant for the otion? What is the displaceent function (t)? Departent of Physics 8

9 (a) rads.68 f Hz.56 T.643 f s (b) c (c) v (9.78)(.).8 s (d) a (9.78) (.).5 s (e) At t =, ( ) cos. v( ) sin sin (f) ( t) cos( t ).cos(9.78t) Departent of Physics 9

10 Eaples At t =, the displaceent of () of the bloc in a linear oscillator is 8.5 c. Its velocity v() then is.9 s, and its acceleration a() is +47. s. What are the angular frequency? What is the phase constant and aplitude? ( t) cos( t ) v( t) sin( t ) a( t) cos( t ) At t =, ( ) cos.85 () v( ) sin.9 () a() cos 47. (3) (3) (): a () a( ) 47. () () 3.5 rads.85 Departent of Physics

11 () (): v() () sin tan cos v() tan ().9 (3.5)(.85).463 o 4.7 or o o o () (): If = 4.7 o, cos.85 cos4.7 o c If = 55 o,.85 o cos c Since is positive, = 55 o and = 9.4 c. Departent of Physics

12 Energy in Siple Haronic Motion Potential energy ( t) cos( t ) U ( t) cos ( t ) Kinetic energy K( t) v( t) sin( t ) v sin ( t ) K( t) / sin ( t ). Departent of Physics

13 Energy in Siple Haronic Motion Mechanical energy E U K cos ( t ) sin ( t ) [cos ( t ) sin ( t )] [cos ( t ) sin ( t )] E U K The echanical energy is conserved! Departent of Physics 3

14 Eaples Suppose the daper of a tall building has ass =.7 5 g and is designed to oscillate at frequency f = Hz and with aplitude = c. (a) What is the total echanical energy E of the daper? (b) What is the speed of the daper when it passes through the equilibriu point? Departent of Physics 4

15 (a) ( f ) (.7 5 )( ).73 9 N The energy: E K U v ( )(.) J 5. MJ (b) Using the conservation of energy, E K U.47 7 v.6 s v (.7 5 ) v Departent of Physics 5

16 An Angular Siple Haronic Oscillator When the suspension wire is twisted through an angle, the torsional pendulu produces a restoring torque given by. is called the torsion constant. Using Newton s law for angular otion, I, I, d. dt I Coparing with the equation of otion for siple haronic otion,. I Since T, T I. Departent of Physics 6

17 Eaple A thin rod whose length is.4 c and whose ass is 35 g is suspended at its idpoint fro a long wire. Its period T a of angular SHM is easured to be.53 s. An irregularly shaped object, which we call X, is then hung fro the sae wire, and its period T b is found to be 4.76 s. What is the rotational inertia of object X about its suspension ais? Departent of Physics 7

18 Rotational inertia of the rod about the center Ia M (.35)(.4) g Since T a Ia and T b Ib Thus Ta T b I I a b Therefore, T b Ib Ta I a (.73 ) 6. g Departent of Physics 8

19 The Siple Pendulu The restoring torque about the point of suspension is = g sin. Using Newton s law for angular otion, = I, gsin, d dt g sin. When the pendulu swings through a sall angle, sin. Therefore d g. dt Coparing with the equation of otion for siple haronic otion, g. Since T, T. g Departent of Physics 9

20 The Physical Pendulu The restoring torque about the point of suspension is = g sin h. Using Newton s law for angular otion, = I, g sin h I, d dt gh sin I. When the pendulu swings through a sall angle, sin. Therefore d dt gh I. Coparing with the equation of otion for siple haronic otion, gh. I Since T, T I. gh Departent of Physics

21 If the ass is concentrated at the center of ass C, such as in the siple pendulu, then T I gh g. g We recover the result for the siple pendulu. Departent of Physics

22 Eaples A eter stic, suspended fro one end, swings as a physical pendulu. (a) What is its period of oscillation T? (b) A siple pendulu oscillates with the sae period as the stic. What is the length of the siple pendulu? Departent of Physics

23 (a) Rotational inertia of a rod about one end M 3 Period T I gh / 3 g / 3g (b) For a siple pendulu of length, T g g 3g c Departent of Physics 3

24 Eaples A physical pendulu has a radius of gyration. When it is suspended at distances l and l fro the center of ass, the periods of oscillation are the sae. (a) Find the relation between l and l. (b) This has been used to deterine g accurately. Find an epression for g. Departent of Physics 4

25 (a) When it is suspended at a distance l fro the center of ass, I M T Ml M. Ml Mgl Siilarly, T ' l' gl' l gl Equating T and T, l l l l ' ' l', l l' l ll' ll' (b) Substituting into the epression of T, T l gl. l l' g g 4 l l' T Departent of Physics 5

26 Eaples A diver steps on the diving board and aes it ove downwards. As the board rebounds bac through the horizontal, she leaps upward and lands on the free end just as the board has copleted.5 oscillations during the leap. (With such tiing, the diver lands when the free end is oving downward with greatest speed. The landing then drives the free end down substantially, and the rebound catapults the diver high into the air.) Modeling the spring board as the rodspring syste (Fig. 5-(d)), what is the required spring constant? Given = g, diver s leaping tie t fl =.6 s. Departent of Physics 6

27 When the board is displaced by an angle, The restoring torque: sin Using Newton s law for angular otion, I 3 3 d dt d 3 dt Coparing with the equation of otion for siple haronic otion, The period should be 3 T 3 T t fl Therefore 48 N 3.6/.5 Departent of Physics 7

28 Daped Siple Haronic Motion The liquid eerts a daping force proportional to the velocity. Then, F d bv, Using Newton s second law, bv a. d dt d b dt b = daping constant.. Solution: ( t) e bt / cos( ' t ), where b 4 '. If b =, reduces to / of the undaped oscillator. If b, then. Departent of Physics 8

29 The aplitude, t e bt / ( ), gradually decreases with tie. The echanical energy decreases eponentially with tie. E( t) e bt /. Departent of Physics 9

30 Eaple For the daped oscillator with = 5 g, = 85 N, and b = 7 gs. (a) What is the period of the otion? (b) How long does it tae for the aplitude of the daped oscillations to drop to half its initial value? (c) How long foes it tae for the echanical energy to drop to half its initial value? (a).5 T.34 s 85 (b) When the aplitude drops by half, Taing logarith, e bt e / bt bt / ln ln t b ln ()(.5)(ln ) s Departent of Physics 3

31 (c) When the energy drops by half, e e bt / bt / Taing logarith, bt ln ln t ln b (.5)(ln ).7.48 s Departent of Physics 3

32 Forced Oscillations and Resonance When a siple haronic oscillator is driven by a periodic eternal force, we have forced oscillations or driven oscillations. Its behavior is deterined by two angular frequencies: () the natural angular frequency / () the angular frequency d of the eternal driving force. The otion of the forced oscillator is given by ( t ) cos( t ). Substituting into the equation of otion, d dt d b dt F cos t, d ( )cos( t ) b sin( t ) F cos t. d d d d d d Using the identity cos( A B) cos Acos B sin Asin B ( ) where d b d cos( t ) d F cos t, cos bd. ( ) b d d F Hence, ( ) b and. d d d Departent of Physics 3

33 () It oscillates at the angular frequency d of the eternal driving force. () Its aplitude. is greatest when d This is called resonance. See Youtube Tacoa Bridge Disaster. Departent of Physics 33

34 34 Departent of Physics ( ), ). ( t i Ae Re t i Be Re, ) ( B A. ) ( A B. B A, B A Two Coupled Oscillators and Noral Coordinates Using Newton s second law, Possible solution, and It is convenient to adopt the third trial solution. Then For non-trivial solutions, we have More generally, we can use the atri for: (A and B are cople.)

35 and for non-trivial solutions,. Either way, we arrive at a secular equation,, or. ( ), If If, A B,, A B. Hence we obtain two solutions. In each solution, the two particles oscillate with the sae frequency. They are called noral odes. Their frequencies are called noral frequencies. Any other solutions are cobinations of the noral odes. Departent of Physics 35

36 Syetric ode: and Acost Antiyetric ode: Acost and Acost In general, the ode that has the highest syetry will have the lowest frequency, while the antisyetric ode has the highest frequency. The syetric ode can be ecited by pulling the two particles fro their equilibriu positions by equal aounts in the sae direction so that ( ) () A and ( ) () The antisyetric ode can be ecited by pulling apart the two particles equally in opposite directions and. then released, so that ( ) () A and ( ) () Departent of Physics 36

37 Eaples Find the frequencies of sall oscillations of a double pendulu. Tangential coponent of forces acting on the upper particle: F t g sin T sin( ) For sall oscillations, T g, sin sin sin( ) Using Newton s second law, g g F t ( ) g g sin ( ) 3g / g / g / g / Departent of Physics 37

38 38 Departent of Physics t i e B A Re / / / / 3 g g g g / / / / 3 g g g g )g / ( A B )g / ( A B et Then For non-trivial solutions, Syetric ode: and Antisyetric ode: and

39 Eaples Consider two pendula of length and ass coupled by a spring with force constant. Find the noral frequencies, the noral odes and the general solution. Syetric ode: g, A = B Antisyetric ode: g, A = B Departent of Physics 39

Simple Harmonic Motion

Simple Harmonic Motion Reading: Chapter 15 Siple Haronic Motion Siple Haronic Motion Frequency f Period T T 1. f Siple haronic otion x ( t) x cos( t ). Aplitude x Phase Angular frequency Since the otion returns to its initial