Oscillations I-Classical Physics-Handouts, Lecture notes of Classical Physics

Download Oscillations I-Classical Physics-Handouts and more Lecture notes Classical Physics in PDF only on Docsity! PHYSICS –PHY101 VU © Copyright Virtual University of Pakistan 41 Summary of Lecture 15 – OSCILLATIONS: I 1. An oscillation is any self-repeating motion. This motion is characterized by: a) The period , which is the time for completing one full cycle. b) The frequency 1/ ,which is the number of T f T= cycles per second. (Another frequently used symbol is ). c) The amplitude A, which is the maximum displacement from equilibrium (or the size of the oscillation). 2. Why does a syste ν m oscillate? It does so because a force is always directed towards a central equilibrium position. In other words, the force always acts to return the object to its equilibrium position. So th restore e object will oscillate around the equilibrium position. The restoring force depends on the displacement , where is the distance away from the equilibrium point, the negative si F k x x= − Δ Δ gn shows that the force acts towards the equilibrium point, and is a constant that gives the strength of the restoring force. k 3. Let us understand these matters in the context of a spring tied to a mass that can move freely over a frictionless surface. The force ( ) (or - because the extension will be ca F x kx k x x= − Δ Δ 2 lled for short). In the first diagram is positive, is negative in the second, and zero in the middle one. The energy 1 stored in the spring, ( ) , is positive in the first and 2 th x x x U x kx= 2 ird diagrams and zero in the middle one. Now we will use Newton's second law to derive a differential euation that describes the motion of the mass: From ( ) and it follows that F x kx dma F m = − = 2 2 2 2 2, or 0 where . This is the equation of motion of a simple harmonic oscillator (SHO) and is seen widely in many different branches of physics. Although we have derived it x d x kkx x dt dt m ω ω= − + = ≡ for the case of a mass and spring, it occurs again and again. The only difference is that , which is called the oscillator frequency, is defined differently depending on the situation. 4. In ω order to solve the SHO equation, we shall first learn how to differentiate some d elementary trignometric functions. So let us first learn how to calculate cos dt starting from the basic defini tω tion of a derivative: Compressed Stretched Relaxed docsity.com PHYSICS –PHY101 VU © Copyright Virtual University of Pakistan 42 ( ) ( ) ( ) ( ) ( ) cos and cos . Take the difference: ( ) cos cos sin sin( / 2) Start : x t t x t t t t x t t x t t t t t t t ω ω ω ω ω ω ω = + Δ = + Δ + Δ − = + Δ − = − Δ + Δ sin as becomes very small. cos sin . (Here you should know that sin for small , easily proved by drawing trian t t t d t t dt ω ω ω ω ω θ θ θ ≈ − Δ Δ ∴ = − ≈ gles.) You should also derive and remember a second important result: sin cos . (Here you should know that cos 1 for small .) 5. What d t t dt ω ω ω θ θ = ≈ ( ) ( ) 2 2 2 2 2 2 happens if you differentiate twice? sin cos sin cos sin cos . So twice differentiating either sin d dt t t dt dt d dt t t dt dt t ω ω ω ω ω ω ω ω ω ω ω = = − = − = − 2 2 2 or cos gives the same function back! 6. Having done all the work above, now you can easily see that any function of the form ( ) cos sin satisfies . But what do , a, b represe t d xx t a t b t x dt ω ω ω ω ω= + = − ( ) nt ? 2 a) The significance of becomes clear if your replace by in either sin 2 or cos . You can see that cos cos 2 cos . That is, the function merely repeats it t t t t t t t πω ω ω πω ω ω π ω ω + ⎛ ⎞+ = + =⎜ ⎟ ⎝ ⎠ self after a time 2 / . So 2 / is really the period of the 2 motion , 2 . The frequency of the oscillator is the number of 1 1 complete vibrations per unit time: so 2 2 mT T k k T m π ω π ω π π ν ω ν ω πν π = = = = = [ ] 1 2 . Sometimes is also called the angular frequency. Note that dim , from it is clear that the unit of is radian/second. b) To understand what and mean let us note that fro k T m T a b π ω ω ω − = = = m ( ) cos sin it d follows that (0) and that ( ) sin cos (at 0). Thus, dt is the initial position, and is the initial velocity divided by . x t a t b t x a x t a t b t b t a b ω ω ω ω ω ω ω ω = + = = − + = = docsity.com