Simple Harmonic Motion Repeated motion with a restoring force that is proportional to the displacement. A pendulum swings back and forth. pendulum A spring bounces up and down. spring
Hooke’s Law When a force is applied to a spring, the displacement is proportional to the force. Change in position changes elastic potential energy. Felastic = -k x Spring force = - (spring constant x displacement)
Sample Problem The springs in a car have a force constant of 9000 N/m. How much will each spring compress if a 450 N passenger enters? F = -k x or 450 N = -(9000 N/m x) x = 450 N / N/m = m
Period and Frequency Simple harmonic motion can be measured as: period – time for one repeated motion period – time for one repeated motion frequency – repeated motions in one second frequency – repeated motions in one second T = period = 1 / frequency f - frequency = 1 / period Example: period T = 0.25 sec, f = 4 Hz
T = period = 1 / frequency f - frequency = 1 / period Example: A grandfather clock pendulum swings back and forth every 2 seconds. What is its period and frequency? 2 seconds is a measure of period Frequency f = 1 / 2 s = 0.5 Hz Period and Frequency
The time for a pendulum to swing back and forth is determined by the length of the cable and the acceleration of gravity. Example: How long must a grandfather clock pendulum cable be to measure 1 second? T = 2 L/g 1 s = 2 L / 9.81 m/s 2 L = 1 s 2 x 9.81 m/s 2 L = 1 s 2 x 9.81 m/s 2 / 4 2 = m Period of a Pendulum Video
The time for a spring to bounce up and down is determined by the force constant of the spring and the mass. Example: If the force constant on a trampoline is 5000 N/m, what is the period of bounce for a 50 kg child? T = 2 m/k T = 2 50 kg / 5000 N/m T = 2s 2 = s T = 2 0.01 s 2 = s Period of a Spring
Types of Waves Mechanical need a physical medium for travel sound, surf, shock waves Electromagnetic can travel through a vacuum light, radio, TV, microwaves
Transverse Waves When the motion of the particles of the medium is perpendicular to the motion of the wave. The wave has crests, troughs, amplitude and wavelength.
Longitudinal Waves When the motion of the particles of the medium is parallel to the motion of the wave. The wave has compressions, rarefactions and wavelengths.
Measurements Of Waves λ = Wavelength = distance between two successive points in phase v = velocity (speed) of a wave
Measurements Of Waves A = Amplitude of a wave The maximum displacement from the equilibrium position
Measurement of waves Wavelength (λ) – distance between two successive points in phase Velocity ( v ) – speed the wave is traveling through the medium Frequency ( f ) – the number of waves passing a point each second v = f x λ or v = λ / T
Example Problem v = f λ or 340 m/s = 256 Hz λ A sound wave travels at 340 m/s. If the frequency of the sound it 256 Hz, what is the wavelength? λ = 340 m/s / 256 Hz = 1.33 m
Amplitude of a Wave Can be the intensity of an earthquake, the loudness of a sound wave or the energy in a tsunami. The maximum displacement from the equilibrium position – proportional to the energy in the wave.
Wave Interference Superposition of waves – two or more waves can exist in the same place at the same time Constructive – waves overlap in phase to increase the amplitude Destructive – waves overlap out of phase to decrease the amplitude Wave A A A A A pppp pppp llll eeee tttt
Standing Waves When two waves of the same frequency and amplitude travel in opposite directions Node – a point where there is complete destructive interference Antinode - a point where there is complete constructive interference
Features of Standing Waves Nodes Points of zero displacement Antinodes Points of maximum displacement
Standing Wave Modes Fundamental (1 ST Harmonic) One Loop L = 1/2 λ 1 st Overtone (2 ND Harmonic) 2 Loops L = 2(1/2) = 1 λ L
Standing Wave Modes 2 nd Overtone (3 rd Harmonic) Three Loops L = 3 λ / 2 3 rd Overtone (4 th Harmonic) 4 Loops L = 4(λ/2) = 2 λ L