Lecture #8-3 Oscillations Siple Haronic Motion So far we have considered two basic types of otion: translation and rotation. But these are not the only two types of otion we can observe in every day life. We can often observe swinging chandeliers surging pistons in engines of cars different pendulus and other siilar exaples. hese all are exaples of oscillatory otion. he ost iportant property of oscillations is that this otion repeats itself with tie. his akes oscillatory otion to be a perfect process to be used as a tie standard. he typical tie quantity associated with oscillations is called period. Period is the tie it takes for one coplete oscillation. In the case when there is no friction or if we can ignore it the period does not change fro one oscillation to another. Since the process is not usually liited by one oscillation only we can also define frequency f as the nuber of oscillations per unit of tie. So f 1. (8.3.1) d i. his eans that SI unit for frequency is s 1 also known as Hertz 1Hz 1s 1 Fro now on we shall liit out attention by a certain type of periodic (oscillatory) otion known as siple haronic otion (SHM). We shall start fro one-diensional exaple of this otion which can be described by the equation b g b g x t x cos t. (8.3.) So the ter siple haronic otion eans that displaceent of a particle-like object changes with tie according to sinusoidal law. here are three constants in the equation x and. Let us find out the physical significance of these constants. he positive quantity x is called aplitude. he aplitude is the axiu displaceent of the particle fro the equilibriu position. Indeed the cosine-function can not have agnitude larger then 1. So the particle s displaceent cannot be larger than x. Actually subscript stands for axiu. Quantity b t g is called the phase of the otion. At tie t 0 this phase is going to be equal. So constant is the original phase. It defines displaceent of the
b g particle at t 0. Indeed x 0 x b g cos so the original phase has to be chosen with accordance to the original displaceent of the particle. Constant is called the angular frequency of siple haronic otion. o understand its significance let us see how it is related to the period of the otion. We know that if is a period then b g b g x t x t x cos t x cos t b b g b g g t t c h t t. Here we have used the fact that cosine-function is -periodic function. hus f. (8.3.3) So the angular frequency is ultiplied by the regular frequency and has units of radians per second. Let us find the velocity of a siple haronic otion. According to definition of velocity we have dx t d vbtg b g cx cosb t gh x sinb t g. (8.3.4) Fro this equation we can see right away that velocity has the aplitude v x. (8.3.5) We can also see that the tie dependence of the velocity is shifted along the tie-axis to the left for one quarter of the period. So the siple haronic otion has the axiu velocity when displaceent is zero and it has zero velocity when at the largest displaceent. his is not surprising because at the farthest point the particle will stop before it changes its direction to the opposite. o find acceleration of a siple haronic otion we have to take one ore derivative with respect to tie which is d x t dv t d abtg b g b g c x b t gh x b t g sin cos. (8.3.6)
So again we can find acceleration s aplitude which is a x. (8.3.7) We can see that this curve is shifted to the left for one quarter of the period copared to the velocity graph. So the displaceent and the acceleration will reach their axiu values at the sae tie but they have opposite directions. When a particle is in the farthest position the acceleration is directed in such a way that it brings it back to the equilibriu. he picture below shows the graphs for all three dependencies in the case when 0. hey all are shown in the sae plot just to see how these quantities are shifted along the axis of tie. In fact they all have different diensions so they should be graphed in separated graphs. he curves are shown by different colors. he red curve is for the displaceent the green one is for the velocity and the yellow is for the acceleration. Equation 8.3.6 shows that b g b g a t d x t b g b g x t x t 0. (8.3.8) he last equation is the second order linear differential equation. he general solution of this equation has the for of 8.3.. Any process which can be described by eans of the equation 8.3.8 represents a siple haronic otion. Let us see an exaple of this process. We shall consider a block of ass which is attached to the spring with spring constant k. his block can ove along one direction which we shall call x-direction. he origin of the x-axis is placed at the block's position where spring is undefored. Now if
we reove this block fro equilibriu for a distance x by copressing or by stretching the spring it will cause the appearance of the force acting on the block fro the spring which will be directed back to the equilibriu. We shall ignore friction so every tie when the block is not in equilibriu the net force acting on it is F kx. According to Newton's second law this force equals a kx a kx 0 d x kx 0 d x k x 0. (8.3.9) he last equation has the sae for as equation 8.3.9 so the block will oscillate with angular frequency k. (8.3.10) he period of this otion will be. (8.3.11) k In fact we have already discussed this behavior. he block which was originally reoved fro the equilibriu will ove back to this equilibriu position. When it passes the equilibriu it has the axiu speed then it oves in the opposite direction for the sae distance x stops there and then goes back. It will continue forever if there is no friction in the syste. In reality however there is always soe friction so these oscillations will be duped and they eventually will stop. Another exaple of the oscillator is the angular siple haronic oscillator. In this case oscillations are also based on the force provided by a spring but it is a different type of the spring. Its springiness is associated with twisting of the suspension wire. his device is called the torsion pendulu where torsion is referred to twisting. It consists of the disk with oent of inertia I suspended fro the wire. he disk can rotate for a sall angle which is counted fro the equilibriu position. Every tie it is reoved fro equilibriu there is the net torque acting on it due to the twisting force of the wire
which brings the disk back to equilibriu. For sall angles this torque is linearly proportional to the angular displaceent which is where is known as the torsion constant so the equation of angular otion of this disk will be I I d d I 0 which is again the equation of siple haronic otion with angular frequency (8.3.1) and period I (8.3.13) I. (8.3.14) Here we can see the sae type of analogy between angular otion and translational otion as we saw earlier. We can go fro one type of otion to the other by replacing x with with I and k with. Let us consider behavior of energy during the siple haronic otion. In fact we have already discussed this behavior when we were talking about conservation of energy (Recall Energy Experients lab). Now let us do it again with the help of atheatical equations. We already know that the energy of the oscillator is transferred back and forth fro the potential energy to the kinetic energy while the su of the two energies which is the total echanical energy of the syste stays the sae (no friction). Potential energy of the spring oscillator is kx t k Ubtg b g x cos b t g. (8.3.15) he kinetic energy of this oscillator is v t k Kbtg b g x b t g x sin sin b t g. (8.3.16) he total echanical energy of oscillator is
b g b g b g b g kx kx kx E U t K t cos t sin t. (8.3.17) hus this energy is conserved since it is always equal to the axiu value of the potential energy when this oscillator was originally reoved fro the equilibriu position. Now we can see that two different eleents of the oscillator are associated with two different fors of echanical energy. he spring stores spring's potential energy while the ass stores kinetic energy. Exercise: Write siilar equations for the energy transforations in angular siple haronic oscillator. Now let us consider pendulus. he potential energy for these oscillators is not elastic potential energy but gravitational potential energy of the earth-pendulu syste. We will start this consideration fro the case of the siple pendulu which consist (see the picture) of the point-like body of ass suspended fro the unstretchable assless string of length L and reoved fro equilibriu for original angle (angular aplitude). We can choose the coordinate syste as it is shown in the picture. Axis y is in the direction of the string and axis x is in the direction tangential to the circular arc along which the body oves. here are two forces acting on the body the tension force directed along the string and the force of gravity g directed downwards. he tension
force has only one coponent along axis y while gravitational force has two coponents in both x and y directions. Let us consider angular otion of this pendulu around the pivot point where the pendulu is suspended. Force of tension does not provide any torque since it is in the sae direction as the radius-vector connecting the body and the pivot point. So the only force having nonzero torque is the force of gravity and the agnitude of this torque equal to Lg sin. So the equation of the angular otion for this pendulu is I Lg sin L d Lg sin d g sin 0. L (8.3.18) Here we have taken into account the fact that for the point like ass I L and the direction of this torque is negative (back to equilibriu). In the case if is a sall (less than 10 degrees) one can write that approxiately sin so the equation becoes like the equation for the siple haronic otion d g 0. L his eans that for a siple haronic otion of the pendulu we have g L L g. (8.3.19) In contrast to the spring this period does not depend on ass of the body. It only depends on the length of the string. You can know recall that you have actually derived these equations fro experiental stand point when you worked on the Pendulus and Springs. lab experient. he first of the equations 8.3.18 is valid not only for a siple pendulu but also for the pendulu of any shape. he object of arbitrary shape suspended fro a pivot point which can perfor oscillatory otion is called a physical pendulu. In the case of
the physical pendulu however one has to use not its length but the distance h fro the pivot point to the center of ass since gravitational force effectively acts at the center of ass. his eans that in the case of the sall angles the equation of otion becoes I d gh 0 d gh I 0. his equation describes siple haronic otion of the physical pendulu with gh I I. gh (8.3.0) (8.3.1) he easureents of the period for the physical pendulu provide the ost efficient way to deterine the value of the acceleration due to gravity. Exaple 8.3.1. Explain how one can find gravitational acceleration based on the easureents of the period of the pendulu ade of a wooden eter stick of length L? he wooden eter stick is the exaple of the unifor rod which has a oent of inertia with respect to the axis of rotation passing through one of its ends I length of this pendulu is h the center of the eter stick so 1 L. he 3 L since the center of ass of the unifor eter stick is at 1 L L Lg g L g g L 3 8 4. 1 3 3 3 We have considered several types of siple haronic otion. hey all are siilar by the fact that displaceent (either the linear displaceent or the angular displaceent) for all these exaples is sinusoidal function of tie. here is also a very interesting siilarity between siple haronic otion and unifor circular otion. Siple haronic otion is just the projection of the unifor circular otion on a diaeter of the circle in which the latter otion occurs. Indeed if we consider the unifor otion of the
particle around the circle with angular speed which starts fro original angle then its projection on axis x is x x cosb t g where x is the radius of the circle. his analogy is also working for the linear speed of the unifor circular otion as well as for its centripetal acceleration. Until this oent we have only considered oscillatory otion without friction. In reality however the oscillatory otion is always daped and eventually stopped by forces of friction. In this case total echanical energy of the syste is not conserved. Part of this energy is spent to perfor work against forces of friction and air resistance so the echanical energy is dissipated to increase teperature of the surroundings. We shall consider the sae exaple as before: the otion of the block on the spring. But now let this otion occur in soe liquid which produces enough resistance or in the air but for the long enough tie so the influence of the air resistance can be detected. If this otion is not very fast we can introduce the daping force which is proportional to the velocity of the block Fd bv where b is known as the daping constant and the inus sign shows that this force is directed opposite to the velocity. So the Newton's second law for this block will give a bv kx a bv kx 0 d x b dx kx 0 d x b dx k x 0 (8.3.) he last equation is the second order linear differential equation of the daped siple haronic otion. he general solution of this equation has a for bt b g b g x t x e cos t. (8.3.3) he sinusoidal ter in this equation represents oscillations while exponential ter shows that these oscillations are daped. his eans that aplitude of oscillations will be decreasing fro its original value x according to exponential law. Substituting solution 8.3.3 back into equation 8.3.4 one can find that the angular frequency of daped siple haronic otion is
k b 4 (8.3.4) which is less than original frequency of the undaped otion. In the case of b 0 when there is no daping this frequency becoes the sae as it was before. In the case of the large enough constant b the angular frequency becoes zero which eans there will be no oscillations at all and the otion will die out during its first period never coing back to the original position. Let us look at the behavior of energy for the daped otion. As we reeber in the case of the undaped oscillations the echanical energy was conserved and its value was E kx. Now it will be decaying during the otion according to the equation E t b g 1 kx e bt So it also decreases exponentially with tie.. (8.3.5) Since in the ost part of cases in the real word daping does exist it is coon practice to use an additional force which provides driven oscillations for the syste. he role of this force is to perfor work which will copensate energy loss due to daping. Usually this force also acts periodically with soe angular frequency d so the syste will perfor the oscillations according to b g b d g x t x cos t. At the sae tie the syste has its own natural frequency. If the driving frequency is equal to the natural frequency d (8.3.6) then the aplitude and velocity of this otion will increase draatically which will course phenoenon known as resonance. In is of extree iportance for the natural frequency of different engineering structures to be different fro the frequencies of external forces acting on the. Otherwise these structures ay collapse under the influence of such periodic forces.