Ch 11 Vibrations and Waves Simple Harmonic Motion Simple Harmonic Motion A vibration (oscillation) back & forth taking the same amount of time for each cycle is periodic. Each vibration has an equilibrium position from which it is somehow disturbed by a given energy source. The disturbance produces a displacement from equilibrium. This is followed by a restoring force. Vibrations transfer energy. Recall Hooke’s Law The restoring force of a spring is proportional to the displacement, x. F = -kx. K is the proportionality constant and we choose the equilibrium position of x = 0. The minus sign reminds us the restoring force is always opposite the displacement, x. F is not constant but varies with position. Acceleration of the mass is not constant therefore. http://www.youtube.com/watch?v=eeYRkW8V7Vg&feature=pl ayer_embedded Key Terms Displacement- distance from equilibrium Amplitude- maximum displacement Cycle- one complete to and fro motion Period (T)- Time for one complete cycle (s) Frequency (f)- number of cycles per second (Hz) * period and frequency are inversely related: T = 1/f f = 1/T Energy in SHOs (Simple Harmonic Oscillators) In stretching or compressing a spring, work is required and potential energy is stored. Elastic PE is given by: PE = ½ kx2 Total mechanical energy E of the mass-spring system = sum of KE + PE E = ½ mv2 + ½ kx2 Here v is velocity of the mass at x position from equilibrium. E remains constant w/o friction. Energy Transformations As a mass oscillates on a spring, the energy changes from PE to KE while the total E remains constant. Displacement, x ranges from –A to + A at the extreme points when the spring is compressed and stretched. At these points when the mass changes direction, v = 0m/s so E = ½ kA2 Thus doubling the amplitude does WHAT to the energy of a system? Period of Oscillation In SHM, the period T of vibration depends on the stiffness of the spring and the suspended mass, but NOT on the Amplitude of oscillation. 2A v 2Af max T 1 2 1 2 kA mvmax 2 2 A m vmax k m T 2 k x = A cos Θ Simple Pendulum A mass oscillating along the arc of a circle with equal amplitude on either side of the equilibrium point resembles simple harmonic motion. Displacement, x = LΘ, is met by a restoring force which would be the component of the weight, mg, tangent to the arc. F = -mgsinΘ. Period of a pendulum If displacement is x = LΘ, then Θ = x/L, so mg F x L mg k L m m T 2 2 k mg/L L T 2 g The mass of a pendulum has no effect on the period. Nor does the amplitude of the swing! So what now? A vibrating object creates a wave. Waves transfer energy from one place to another. Waves travel with or without mediums. A medium can be solid, liquid, or gas, but it affects how fast a wave travels. Some waves like light travel without a medium. These are called Electromagnetic Waves. All waves have a wavelength (λ), a frequency (f), a period (T), and an amplitude (A or x). Wave diagrams Here is a common wave model: F v T m /L The medium determines the wave velocity. In a string, Tension also effects velocity since particles are held close together. Forced Vibrations All objects have a Natural Frequency at which they vibrate at when set into motion. An object can be set into vibration at a different frequency other than it’s natural frequency, f0. This is called a forced vibration, f. The amplitude of this vibration depends on the difference between f and f0. When f = f0 the amplitude is max and often dramatic. This is called the Resonant Frequency. Resonance occurs when an object is forced into vibrations at it’s natural frequency. http://www.youtube.com/watch?v=Az503VJ6kHw&feature=player_ detailpage Tacoma Narrows Wave Motion A wave is not matter but energy moving through matter (or through a vacuum) A single bump is a pulse. A continuous or periodic wave is a repeating disturbance caused by a vibrating source. The frequency of the wave matches the frequency of the source. Wave crests travel a distance of one wavelength λ in a time of one period T. Thus velocity v=λ/T. Since f=1/T, then v = f λ. This is the wave equation. Wave Types Longitudinal: Medium moves parallel to wave energy. Examples include sound and slinky B(bulk mod ulus) v velocity depends on elasticity & inertia of (density ) medium Transverse: Medium moves perpendicular to wave energy. Most waves are this type including light, water, and Electromagnetic waves Surface: Currents in the surface of the ocean sometimes travel in small circles. These are combinations of the above. http://www.acs.psu.edu/drussell/Demos/waves/wavemotion.html Your turn to Practice Please do the HW problems on pgs 323 & 324 #s 1-7, 28 & 30. Complete the Ch 11 Reading worksheet if you have not already done so. Wave Behavior When a wave pulse strikes an obstacle or barrier, at least part of the wave is reflected. Law of Reflection Wave fronts are all points along a wave forming the crest. Ray- a line perpendicular to the wave front For every ray striking a surface and being reflected back into the original medium, the Law of Reflection holds true. “Normal” is perpendicular to surface Principle of Superposition Unlike matter, two or more waves can pass through the exact same space at the same time. In the area where the waves overlap, the resulting displacement is the algebraic sum of their individual displacements. This is the Principle of Superposition Standing wave formations Interference The position of wave crests relative to each other is described by Phases: Waves are IN PHASE when crests and troughs are aligned and the result is CONSTRUCTIVE INTERFERENCE. Waves are OUT OF PHASE when crests and troughs produce DESTRUCTIVE INTERFERENCE. Standing Waves When an incident wave and a reflective wave interfere at just the right frequency, they can form a standing wave. This wave appears to have fixed nodes (zero displacement) and antinodes (maximum amplitude). Standing waves are produced when objects vibrate at their natural frequency or resonant frequency. Harmonics & Overtones The wavelengths of standing waves are directly related to the length of the string, L. The fundamental frequency, lowest resonant frequency, is one loop. This L = ½ λ1. (AKA first harmonic) The other natural frequencies are called overtones. A whole # multiple of the fundamental. These are also called harmonics. st Second harmonic has 2 loops. L = λ2 (recall λ1 is the 1 harmonic). Any harmonic n can be found by n where n=1,2,3,… L n 2 Think of n as # of loops. Your turn to Practice Please do the remaining HW problems on your Reading worksheet pgs 342-345. .