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Phys2212_20.1+to+20.7

Phys2212_20.1+to+20.7 - Physics 2212 Waves Lecture 1...

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Physics 2212 Waves Lecture 1 Traveling Waves

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April 18, 2005 Physics 2212 - Lecture 1 Example : A Damped Pendulum A 500 g mass on a 50 cm long string oscillates as a pendulum. The amplitude of the pendulum is observed to decay to ½ of its initial value after 35 s. (a) What is the time constant τ of the damped oscillator? (b) At what time t 1/2 will the energy of the system have decayed to ½ of its initial value? max max At 0, , and at 35 s, / 2. t x A t x A = = = = / 2 max / 2 / 2 ln 2 t x Ae A t τ - = = = / 2ln 2 (35 s) / 2ln 2 25.2 s t τ= = = 1/ 2 / 0 0 1/ 2 / 2 / ln 2 t E e E t - = = 1/ 2 ln 2 (25.2 s)ln 2 17.5 s t = = = m b τ≡ / 2 0 ( ) cos( ); t x t A e t ϖ φ - = + 2 2 4 k b m m ϖ= - 2
April 18, 2005 Physics 2212 - Lecture 1 Driven Oscillations 2 2 cos d d F d x b dx k x t dt m dt m m ϖ + + = Now, suppose we drive a damped mechanical oscillator with an external force F(t) = F d cos( ϖ d t ) . Then the equation of motion is: The system will show the property of resonance The oscillation amplitude will depend on the driving frequency ϖ d , and will have its maximum value when: i.e., when the system is driven at its resonant frequency ϖ 0 . 0 / d k m = = 3

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April 18, 2005 Physics 2212 - Lecture 1 Resonance When ϖ 2 = k/m , the first term in G vanishes and the amplitude of the oscillation is a maximum. This is the resonance condition. The width of the resonance curves depends on b , i.e., on the amount of damping. Wider curves with smaller resonance curves correspond to more damping and larger values of b . 2 2 cos d d F d x b dx k x t dt m dt m m ϖ + + = The solution of this equation for driven oscillations is: cos( / 2), where d F x t G φ π = - + 2 2 2 ( ) ( ) ; G m k b = - + 1 2 tan b m k - = - 4
April 18, 2005 Physics 2212 - Lecture 1 The Wave Model We will focus on the basic properties of waves using the wave model , which emphasizes the aspects on wave behavior common to all waves (e.g., water waves, sound waves, light waves, etc.) The wave model is built around the idea of traveling waves, wave disturbances that travel with a well-defined speed. We will begin by distinguishing three types of waves: 1. Mechanical waves can travel only within a medium, such as air or water. Examples: sound waves, water waves. 1. Electromagnetic waves are self-sustaining oscillations that require no medium and can travel through a vacuum. Examples: radio waves, microwaves, light, x-rays, gamma rays, etc. 1. Matter waves also can travel in vacuum and are the basis for quantum physics (i.e. quantum mechanics). Examples: quantum wave functions for electrons, photons, atoms, etc. 5

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April 18, 2005 Physics 2212 - Lecture 1 Two Types of Wave Motion A transverse wave is a wave in which the particles of the medium move perpendicular to the direction of wave motion. They can be polarized. Examples: waves on a string, electromagnetic waves. longitudinal wave parallel to the direction of wave motion. They cannot be polarized.
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Phys2212_20.1+to+20.7 - Physics 2212 Waves Lecture 1...

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