Phys2212_20.1+to+20.7

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

Info iconThis preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
Physics 2212 Waves Lecture 1 Traveling Waves
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 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
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 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
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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.
Background image of page 6
Image of page 7
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 51

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

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online