OSCILLATIONS If any quantity changes regularly with time about a fixed point it can be said to be oscillating. displacement time displacement time If the displacement-time graph (or v-t or a-t graphs) is sinusoidal then the oscillator is said to be oscillating with Simple Harmonic Motion. Simple harmonic motion A body undergoing SHM oscillates about a fixed point in a sinusoidal pattern and experiences a single force (hence an acceleration) which is directly proportional, and opposite, to the displacement. fixed point xxx xx F FF The fixed point is the equilibrium position, i.e. the position at which the body would come to rest if it were to lose all of its energy. The amplitude of the motion is the maximum displacement from the equilibrium position. Any system that undergoes SHM exhibits two key features: when the system is displaced from equilibrium there must exist a restoring force that tends to restore it to equilibrium the restoring force must be proportional to the displacement fixed point xxx xx F FF The restoring force, F = - kx where k is a constant fixed point F FF x = max x = 0 x = max v = 0 ms -1 v = max F = maxF = 0 NF = max a = maxa = 0 ms -2 a = max Mass on a spring The mass m attached to a spring exhibits shm. The period of oscillation is T = 2 m/kwhere k is the spring constant The period is independent of the amplitude. Such oscillations are known as isochronous. Mass on a pendulum The period of oscillation is T = 2 l/gwhere l is the length of the spring The period of oscillation for a pendulum is independent of the amplitude and the mass of the pendulum. SHM is related to circular motion. = t O Consider the horizontal displacement of the object, xx x = r cos r x = r cos t For a pendulum bob, x = A cos t where A = amplitude SHM mathematics F = - kx F = ma ma = -kx a = -k x m x = A cos t a = -k A cos t m x = sin dx = cos dt x = cos dx = - sin dt x = sin a dx = a cos a dt x = cos a dx = - a sin a dt a = any constant x = A cos t dx = - A sin t dt v = - A sin t dv = dt d 2 x = dt 2 - A 2 cos t= - 2 x a = = t Oxx r So this is SHM (a x). SHM equations x = A cos t Displacement Velocity v = -A sint Acceleration a = -A 2 cos ta = - 2 x From Pythagoras,1 = sin 2 + cos 2 Rearranging,sin= 1 - cos 2 sin t = 1 cos 2 t Recall v equation,v = A 1 cos 2 t v = A 2 A 2 cos 2 t v = A 2 x 2 v is max when x=o SHM equations x = A cos t Displacement Velocity v = -A sint Acceleration a = -A 2 cos ta = - 2 x v = A 2 x 2 x = A sin t Displacement Velocity v = A cost Acceleration a = -A 2 sin ta = - 2 x v = A 2 x 2 At t=0, x = A At t=0, x = 0 We have seen previously that a = -k x m So, -k x m = - 2 xk = 2 m SHM graphs SHM energy Energy of a swinging pendulum KE = 0 KE = max GPE = max GPE = 0 The energy of the pendulum is continuously changing from KE to GPE. The total energy of the system remains constant, assuming there are no energy losses. So at any point of its motion, The energy of the pendulum is continuously changing from KE to GPE. The total energy of the system remains constant, assuming there are no energy losses. Total Energy = KE + GPE = 1/2mv 2 + mgh KE = 0 KE = max GPE = max GPE = 0 Try and derive and expression for KE of a SHM oscillator, using the SHM equations. Using v = A cost, KE = m v 2 KE = m A 2 2 cos 2 t Ke max = m A 2 2 Total E = PE + KE So, PE = Total E - KE PE = m A 2 2 - m A 2 2 cos 2 t PE = m A 2 2 ( 1 - cos 2 t) PE = m A 2 2 sin 2 t Energy of a Mass on a spring x = 0 x = +Ax = -A KE = 0 KE = maxKE = 0 PE = max PE = 0 Total Energy = KE + PE = 1/2mv 2 + kx 2 where 1/2kx 2 is the spring potential energy Damped and forced oscillations An oscillator that is set into motion and left alone will oscillate at its own frequency. This is the oscillator s natural frequency. When the oscillator is being forced to oscillate at a different frequency by an external driving force, then it is said to be executing forced oscillations. The frequency the oscillator is forced to vibrate at is known as the driving frequency. Damped vibrations Unless an oscillator is maintained by some source of energy, its amplitude of vibration will become progressively smaller - the motion is said to be damped. The decrease in amplitude occurs because some of the energy of the oscillating system is used to overcome resistive forces. The greater the resistive forces, the greater the damping, i.e the amplitude of vibration will be reduced much quicker. Underdamped With light damping gradually reduces in amplitude but takes a long time to disappear. Overdamped With heavy damping the oscillator might not even complete one cycle or if does return to its equilibrium position, it will take a very long time. Critically damped With critical damping the oscillator returns as quickly as possible to the equilibrium without overshooting (going past the equilibrium position). Resonance When the driving frequency is the equal to the natural frequency of the system that is being driven, it oscillates with maximum frequency. This is called resonance. Examples of resonance? washing machine vibrating dashboard smashing wine glass with sound collapse of Tacoma Narrows Bridge collapse of Angers bridge The graph shows the amplitude with which a simple harmonic oscillator will vibrate, against the driving frequency applied to it. driving frequency / Hz amplitude / m f0f0 Phase The phase is related to the position along the sine curve. The phase difference is measured as an angle. A complete cycle is 2 radians. Phase differenceIn radians A quarter of a cycle Half a cycle cycle A full cycle /2 3/4 2 Wave characteristics Mechanical waves require a medium to travel through. e.g sound waves, water waves, seismic waves (earthquakes) Electromagnetic waves require no medium through which to travel, they can travel through a vacuum (empty space) e.g microwaves, radio waves, X-rays etc. What is a wave? A wave is a periodic disturbance through a medium or space. It is caused by a vibrating source and travels outwards from the source. Waves transfer energy from one place to another, without transferring matter. Waves that travel out of a source or move from one place to another are called progressive waves or travelling waves. There are two types of waves: Transverse The particles oscillate perpendicular to the direction of propagation of the wave. Direction of propagation of wave Longitudinal The particles oscillate parallel to the direction of propagation of the wave. rarefactions compressions Circular waves wavefronts rays (these are lines joining points of the wave that are all in phase) Wave quantities Equilibrium position (point at which particles remain in place when there is no disturbance) crest trough Wavelength, Wavelength, : the distance between two adjacent identical points on the wave. (meters, m) amplitude Amplitude: the maximum displacement of any point on the wave from its equilibrium position. Frequency, f : the number of waves produced or pass a point in one second. (hertz, Hz) Period, T: the time it takes for one wave to be produced, or to pass a point. If f = 10 Hz (10 cycles/second), what is T? T = 1/10 = 0.1 s Frequency, f = 1 Period, T Speed of wave, v = distance travelled by wave / time taken v = T v = f the wave equation Intensity, I I A 2 Graphical representation of waves Displacement - position Displacement Position Wave snapshot Displacement - time Displacement Time T Wave properties Reflection Refract Diffract Interfere (superposition) Reflection Plane waves normal The laws of reflection: angle of incidence = angle of reflection i = r the incident and reflected rays are in the same plane as the normal. Reflection of a wave pulse String fixed on support fixed end string pulls support upwards support pulls string downwards Refraction Change of direction due to change of speed. Waves are refracted when they travel from one medium to another. transmitted sin i sin r = v1v1 v2v2 Snells law: sin i sin r = n where n = refractive index The refractive index of a medium is a measure of how much the speed of a wave is reduced in that medium. Diffraction Waves spread out when they go through an opening which is comparable to their wavelength. Not much diffraction gap is much bigger than the wavelength. A lot of diffraction wave is similar size to wavelength Interference When two or more waves meet at a point the total displacement is the vector sum of the displacements of the individual waves. This is the Principle of Superposition. superposition = placing on top The pulses add up when they cross. The pulses cancel out (totally if magnitude is the same). CONSTRUCTIVE INTERFERENCE DESTRUCTIVE INTERFERENCE Double slit interference pattern wave crestswave troughs Interference maxima Waves are in phase constructive interference. Path difference = 0 Interference minima Waves are out of phase destructive interference. Path difference = \ Same wavelength, dippers are twice as far apart. Constructive interference occurs whenever the path difference between coherent sources is n, where n is a whole number. Destructive interference occurs whenever the path difference between coherent sources is n / 2, where n is a whole number.