Pre-AP Physics – Chapter 14 Notes – Yockers Vibrations and Waves Hooke’s Law Fs kx →Fs = force of a spring (may be referred to as a “restoring force”) →k = spring constant (reflects the stiffness of the spring) →x = displacement (x = 0 is the equilibrium position) - the direction of the restoring force (Fs) is such that the object is being either pushed or pulled toward the equilibrium position - what is the description of an object’s motion in a system governed by Hooke’s law? →Fs →v and p →a - simple harmonic motion occurs when the net force along the direction of motion is a Hooke’s law type of force – that is, when the net force is proportional to the displacement and in the opposite direction - the energy stored in a stretched or compressed spring or other elastic material is called elastic potential energy (PEs) 1 PE kx 2 s 2 - energy is stored in a spring only when it is either stretched or compressed - the elastic potential energy is a maximum when a spring has reached its maximum compression or extension - PEs is always positive Simple Pendulum - restoring force is a component of the bob’s weight - the motion of a pendulum is not SHM - if the angular displacement from equilibrium is small (less than about 15 degrees, so sin=) then a pendulum follows the general equation form of Hooke’s law, and it is justified to say there is SHM Amplitude, Period, and Frequency - amplitude (A) ≡ the magnitude of the maximum position of the object relative to its equilibrium position - period (T) ≡ the time required for one complete vibration (one complete cycle of motion) - frequency (f) ≡ the number of vibrations (oscillations) per second (also the inverse of T) →SI unit of frequency is the Hz or s-1 1 1 f or T T f - the period of a pendulum is given by L T 2 g →L = fixed length of the pendulum →g = the acceleration due to gravity (not necessarily the Earth’s!) →period does not depend upon the mass of the bob →period does not depend upon amplitude - how can a time piece using a pendulum be adjusted if it runs fast/slow? - the period (T) of a shadow of an object moving with uniform circular motion is equivalent to the horizontal motion of an object on the end of a spring, so m T 2 k →period depends upon the mass of the oscillating object and the stiffness of the spring →period does not depend upon amplitude of the oscillation Properties of Waves - wave – motion of a disturbance - medium – material through which a disturbance travels →particles within a wave vibrate around an equilibrium position, and changes in their vibration allow the net disturbance to travel from one location to another →mediums – air, water, ground …. →each point in a medium can be treated as a simple harmonic oscillator vibrating with the same frequency as the disturbance causing the wave - a mechanical wave’s propagation requires the existence of a medium (electromagnetic waves do not require a medium) - wave types →pulse – a single nonperiodic disturbance →periodic – source is some form of periodic motion - sine waves describe particles vibrating with simple harmonic motion - transverse wave – particles vibrate perpendicularly to the direction of wave motion →crest, trough →wavelength () – distance between two adjacent similar points of the wave - longitudinal wave – particles vibrate parallel to the direction of wave motion →can also be described by a sine wave - density wave or - pressure wave - superposition – displacement is algebraic sum of displacements of individual wave pulses Period, Frequency, and Wave Speed - when vibrating particles of a medium complete one full cycle, one complete wavelength passes any given point - frequency describes the number of crests that pass a given point in a unit of time - period is the amount of time required for one complete vibration of the particles of the medium - speed of a wave x v t v f T - the speed of a wave changes only when the waves move from one medium to another or when properties of the medium are changed Wave Interference - superposition principle – if two or more traveling waves are moving through a medium, the resultant wave is found by adding together the displacements of the individual waves point by point) - constructive interference – when waves meet crest to crest and trough to trough, the superposition principle results in a greater amplitude/displacement - destructive interference – when waves meet crest to trough, the superposition principle results in a smaller amplitude/displacement Reflection - whenever a traveling wave reaches a boundary between mediums, part or all of the wave is reflected - "fixed" reflection – reflection of a pulse traveling on string that is fixed at one end →pulse is inverted →string exerts an upward force, wall exerts a downward force →this downward force causes the pulse to be inverted when reflected - "open" reflection – reflection of a pulse traveling on a string that is attached to a ring of negligible mass that is free to slide along a post without friction →pulse is not inverted →upon reaching the post, the pulse exerts a force on the ring →the ring accelerates upward →ring returns to its original position by the downward component of the tension - when a pulse reaches a boundary, it is partly reflected and partly transmitted past the boundary into the new medium Standing Waves - standing wave – wave pattern that results when two waves of the same frequency, wavelength, and amplitude travel in opposite directions and interfere (wavelengths that do not allow both ends to be nodes do not produce standing waves) - standing waves can be set up in a stretched spring by fixing one end to a stationary object and connecting the other end to a vibrating object →traveling waves reflect from the fixed end, creating waves traveling in both directions →the incident and reflected waves combine according to the superposition principle →if the string is vibrated at the right frequency, the wave appears to stand still - node – occurs where the two traveling waves always have the same magnitude of displacement but have opposite sign, so that the net displacement is zero at that point (“no motion”) - antinode – midway between nodes, where the string vibrates with the largest amplitude - all points on the string oscillate together vertically with the same frequency but that different points have different amplitudes of motion →nodes are stationary, and the points at which the string is attached to the →stationary object and vibrating object are also nodes →the distance between nodes is equal to one half of the wavelength of the wave 1 d NN 2 .