f 1. (8.1.1) This means that SI unit for frequency is going to be s 1 also known as Hertz d1hz

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1 ecture 8-1 Oscillations 1. Oscillations Simple Harmonic Motion So far we have considered two basic types of motion: translational motion and rotational motion. But these are not the only types of motion in nature. We can often see swinin chandeliers surin pistons of enines various pendulums. hese are all examples of oscillatory motion. he most important feature of oscillations is that this motion repeats itself with time. his makes oscillatory motion to be a perfect process to use for time standard. he standard time quantity associated with oscillations is called the period. he period is the time required for one complete oscillation. If there is no friction or if we can inore friction the period does not chane from one oscillation to another. Since the process usually is not limited by one oscillation only we can also define frequency f as the number of oscillations per unit of time. So f 1. (8.1.1) his means that SI unit for frequency is oin to be s 1 also known as Hertz d1hz 1s 1 i. We shall limit our attention by a certain type of periodic (oscillatory) motion known as simple harmonic motion (SHM). et us start from one-dimensional example of this motion which is x t Acos t. (8.1.) So the term simple harmonic motion means that displacement of a particle-like object is described by sinusoidal function of time. here are three constants in equation 8.1. A and. et us determine what the physical sinificance of these constants is. he positive quantity A is called amplitude. he amplitude is the maximum displacement of the particle from its equilibrium position. Indeed cosine function cannot have manitude larer than 1. So the particle s displacement cannot be larer than A. he quantity b t is called phase of the motion. At time t 0 this phase is equal. So constant is the oriinal phase. It defines the displacement of the particle at

2 t 0. Indeed x 0 oriinal displacement of the particle. Acos so the phase constant has to be chosen accordin to Constant is called the anular frequency of the simple harmonic motion. o understand its meanin let us see how it is related to the period of motion. We know that if is the period then c x t x t Acos t Acos t t t t t. Here we used the fact that cosine function is periodic function. hus h f. (8.1.3) So the anular frequency is times the reular frequency and has units of radians per second. et us determine velocity of the simple harmonic motion. o do so we can use the analoy between simple harmonic motion (SHM) and uniform circular motion. Onedimensional SHM can be considered as projection of the radius-vector for the uniform circular motion on one direction. We know that for the uniform circular motion linear velocity and radius are related as v r. akin into account the fact that linear velocity (tanential to the circle) is perpendicular to the radius and considerin its projection on the same direction we have v t A sin t. (8.1.4) From this equation we can see that velocity has the amplitude of vm A. (8.1.5) We can also see that time dependence of velocity is shifted alon the axis of time to the left for one quarter of the period. So the simple harmonic motion has the maximum velocity when displacement is zero and it has zero velocity at the larest displacement. his is no wonder because in the farthest point the particle will stop before it chanes its direction to the opposite.

3 o find acceleration of the simple harmonic motion we shall use the same method considerin projection of the circular motion on one direction. We have to take into account that centripetal acceleration is a to the center of the circle so we have r and it is directed alon the radius vector a t Acos t. (8.1.6) Aain we can find acceleration amplitude from here which is am A. (8.1.7) We can see that this curve is shifted to the left for one quarter of the period compared to the velocity raph. So displacement and acceleration will reach their maximum values at the same time but they have opposite directions. When the particle is in the farthest position the acceleration is directed in such a way that it brins the particle back to equilibrium. he picture above shows the raphs for all three dependencies in the case when 0. We showed them all at the same raph just to see how these quantities are shifted alon the axis of time. In fact they all have different dimensions so they should be raphed on separated raphs. his is why there are no units shown. he curves are shown by different colors. he red curve is for displacement the reen one is for velocity and the yellow is for acceleration. Equation shows that a t x t. (8.1.8) As an example of oscillations let us consider the block of mass m which is attached to a sprin with sprin constant k. his block can move in one direction which we will

4 call x-direction. he oriin of axis x is placed at the block's position when sprin is undeformed. If we remove this block from equilibrium for distance A by compressin or stretchin the sprin it will be force actin on the block from the sprin which will be directed in such a way that it brins the block back to equilibrium. For now we shall inore friction so every time when the block is not in equilibrium the net force actin on it is F kx. Accordin to Newton's second law we have ma kx ma kx 0 m( x) kx 0 k x m x 0. he last equation shows that the block will oscillate with anular frequency (8.1.9) k m. (8.1.10) he period of this motion will be m. (8.1.11) k We have already proved this equation experimentally in the lab. he block which is oriinally removed from its equilibrium will move back to the equilibrium position. When it passes equilibrium point it has the maximum speed then it oes on the opposite side of equilibrium point for the same distance A stops there and then oes back. It will continue forever if there is no friction in the system. In reality however always there is some friction so oscillations will be dumped and eventually will stop. et us see the behavior of enery durin simple harmonic motion. We already know that the enery of the oscillator transfers back and forth from potential enery to kinetic enery while the sum of the two the total mechanical enery stays the same (no friction). Potential enery of the sprin oscillator is kx t k PE t A cos t (8.1.1) he kinetic enery of this oscillator is

5 mv t m k sin KE t A sin t A t (8.1.13) he total mechanical enery of oscillator is E PE t KE t k A cos t k A sin t k A. (8.1.14) hus this enery is conserved since it is always equal to the maximum value of potential enery when this oscillator was removed from equilibrium position. Now we see that the two different elements of the oscillator are associated with two different forms of mechanical enery. he sprin stores sprin's potential enery while the mass stores kinetic enery. et us now consider the most well-known type of oscillators the pendulum. he potential enery for these oscillations is not the potential enery of a sprin but ravitational potential enery of the Earth-pendulum system. We shall start from the case of the simple pendulum which consist (see the picture) of the point-like body of mass m suspended on unstretchable massless strin of lenth and removed from equilibrium for oriinal anle (anular amplitude) m. We choose the coordinate system as it is shown in the picture. Axis y is in the direction of the strin and axis x is in the tanential direction to the circular arc alon which the body moves. here are two forces actin on the body the tension force directed alon the strin and the force of ravity m directed downwards. he tension force has only one component alon axis y while ravitational force has two components in both x and y directions. et us consider the anular motion of this pendulum. Force of

6 tension does not provide any torque since it is in the same direction as the radius-vector connectin the body and the pivot point. So the only force havin nonzero torque is the force of ravity and the manitude of this torque equal m sin. he equation of the anular motion for this pendulum is I m sin sin 0. (8.1.15) Here we took into account that for the point-like mass I m and the direction of torque is neative (back to equilibrium). In the case if is a small anle (less than 10 derees) one can assume that sin. So the equation becomes like the equation of the simple harmonic motion a x 0 which is a 0. his means that for the simple harmonic motion of the pendulum we have. (8.1.16) In contrast to a sprin this period does not depend on mass of the body. It only depends on the lenth of the strin. he first of equations is valid not only for the simple pendulum but for the pendulum of any shape. he object of arbitrary shape suspended from the pivot point which can perform oscillatory motion is called a physical pendulum. In the case of the physical pendulum one has to use not its lenth but the distance h from the pivot point to the center of mass since ravitational force is applied at the center of mass. his means that in the case of the small anles the equation of motion becomes I mh 0 mh I 0. (8.1.17) his equation describes simple harmonic motion of the physical pendulum with

7 mh I I. mh (8.1.18) he measurements of the period for the physical pendulum provide the most effective way to obtain the value of the acceleration due to ravity. Example How to find ravitational acceleration based on measurements of the period of the pendulum made of the wooden meter stick of lenth? he wooden meter stick is the example of the uniform rod which has a moment of inertia with respect to the axis of rotation passin throuh one of its ends I lenth of this pendulum is h the center of the meter stick so 1 m. he 3 since the center of mass of the uniform meter stick is at 1 m m

OSCILLATIONS

OSCILLATIONS OSCIAIONS Important Points:. Simple Harmonic Motion: a) he acceleration is directly proportional to the displacement of the body from the fixed point and it is always directed towards the fixed point in