What do these two systems have in common? Gravity is the driving force If it is frictionless, energy is conserved It would continue indefinitely

We know springs have potential energy when they are stretched or compressed U = ½ kx 2 They have potential energy because of the restoring force of the spring F by spring = -kx (Hooke’s Law) Restoring force is negative because it is in the opposite direction of the displacement - F - 2F

A family of four with a total mass of 200 kg step into their car causing the springs to compress 3.0 cm. What is the spring constant of the springs assuming they act as a single spring? How far would the car lower if loaded with 300 kg rather than 200 kg?

Consider a spring/block system where the spring is stretched to some displacement “x” Restoring force returns block to its natural position, but the block continues to move to a compressed position “x” because of its inertia If the system is frictionless, the block would continue to oscillate indefinitely. It would demonstrate simple harmonic motion.

SHM Terms and Concepts Amplitude (A) – the maximum displacement away from the equilibrium point (m) Cycle – one complete motion forward and back Period (T) – the time it takes to complete one complete cycle (s) Frequency (f) – the number of cycles per second (Hz, s -1 ) Maximum velocity (v max ) – the velocity at the equilibrium position

A spring stretches m when a kg mass is gently lowered on it. If the system is then placed on a frictionless table and the mass is pulled m from its equilibrium position, determine (a) the spring constant, (b) the maximum speed attained during an oscillation, (c) the speed when the mass is m from equilibrium, and (d) the maximum acceleration of the mass.

Relationship between UCM and SHM SHM is related to UCM in that it is UCM observed from a different perspective. SHM is also related to wave motion

Mechanical energy is conserved U max = K max ½kA 2 = ½mv max 2 A 2 / v max 2 = m / k A / v max = √(m / k) Period of oscillating springs affected by mass and spring constant… NOT the amplitude. The period in SHM is analogous to the period in UCM v max = 2πA / T A / v max = T / 2π T = 2 π √(m / k) Oscillating Springs What variables in the block/spring system would affect the period of oscillation?

A spider of mass 0.30 g waits in its web of negligible mass. A slight movement causes the web to vibrate with a frequency of 15 Hz. (a) What is the value of k for the elastic web? (b) At what frequency would you expect the web to vibrate if an insect of mass 0.10 g were trapped in addition to the spider?

Pendulums Two forces act on a pendulum with length L – the weight (mg) and the tension (F T ) The force that causes the pendulum to move is tangent to the arc of movement: F = -mgsinθ For small angles, sin θ = θ F = -mgθ What determines the period of a pendulum? Since θ = x / L F = -mgx/L

-F/x = mg/L At small angles, a pendulum exhibits SHM. SHM period equation: T = 2π√(m/k) Hooke’s Law: F = -kx, k = -F/x k = mg/L T = 2π√(m/(mg/L)) T = 2π√(L/g) Period of pendulum determined by length and gravity

A geologist uses a simple pendulum that has a length of cm and a frequency of Hz at a certain location on the Earth. What is the acceleration due to gravity at this location?

The length of a simple pendulum is 0.5 m, the pendulum bob has a mass of 25 g, and it is released so that the bob is 10 cm above its equilibrium position. (a) With what frequency does it oscillate? (b) What is the pendulum bob’s speed when it passes through the lowest point of the swing?

Wave Motion p/~norimari/science/Jav aEd/e-wave1.html Objects that vibrate in SHM can produce periodic waves which share many of the same characteristics as SHM objects. Mechanical wave – a disturbance that travels through some type of medium or substance Ex: Seismic waves Ocean waves Sound waves Nonmechanical wave – a disturbance that does not require a medium or substance to travel Ex: Electromagnetic radiation (light)

Transverse (shear) waves cause particles to move back and forth at right angles to the direction of wave travel Longitudinal (compressional) waves cause particles to move back and forth in the same direction as wave travel (SOUND is this type)

A few characteristics about transverse waves: Crest – the high point of a wave (the “peak”) Trough – the low point of a wave (the “valley”) Wavelength – the distance between two successive crests (or troughs) in a wave Amplitude – the distance from the midpoint to the crest (or the trough) of a wave Frequency – the number of wavelengths that pass a given point in one second Period – the amount of time it takes for one wavelength to pass a given point

The speed of a wave is equal to the frequency times the wavelength. Speed of sound in air  v = ( T) m/s T = temp (C) Speed of light  v = 3.00 x 10 8 m/s Speed of wave in cord  v = √F T / (m/L) F t = tension m = mass L = length Typical Wave Speeds V = f λ v = speed (m/s) f = frequency (Hz, s-1) λ = wavelength (m)

A periodic wave has a wavelength of 0.50 m and a speed of 20 m/s. What is the wave frequency? Radio waves from an FM station have a frequency of MHz. If the waves travel with a speed of 3.00 x 10 8 m/s, what is the wavelength?

Sound as a Wave Sound is produced by a vibrating object which creates a disturbance of air molecules in the form of a longitudinal wave. Path of vibration air  eardrum  three bones  cochlear window  cochlear fluid

Resonance The frequency determined using the SHM formulas is called the natural frequency. 1 / f = 2 π √(m / k) 1 / f = 2 π √(L / g) The system vibrates at this frequency on its own. Without friction, it would vibrate indefinitely. When an externally applied frequency matches an object’s natural frequency, the amplitude of the vibration gets larger. This phenomenon is called resonance. Example: Child being pushed on swing Marching in step across bridge Breaking glass with voice

Resonance in Waves Resonance also appears when repeated waves at a certain frequency interfere with each other. Interfering waves obey the principle of superposition The amplitude of the resultant wave is the sum of the amplitudes of the individual waves

Situation (b) is called constructive interference because the two individual waves combine to make a larger wave. Situation (a) is called destructive interference because the two individual waves cancel each other out.

Constructive and destructive interference can occur simultaneously. Where a crest meets a crest (constructive), the waves are in phase. Where a crest meets a trough (destructive), the waves are out of phase.

When periodic waves are created in a cord, interference occurs between initial and reflected waves. When the frequency of waves matches the cord’s natural frequency, a standing wave is produced. Standing waves do not appear to be traveling, but rather appear to oscillate. They have nodes (points of destructive interference) and antinodes (points of constructive interference)

Fundamental (1 st Harmonic) L = (1/2)λ f1f1 First Overtone (2 nd Harmonic) L = λ f 2 = 2f 1 Second Overtone (3 rd Harmonic) L = (3/2)λ f 3 = 3f 1

The velocity of waves on a string is 92 m/s. If the frequency of the 1 st overtone is 475 Hz, how far apart are two adjacent nodes?

Resonance in Musical Instruments Wind instruments resonate when standing waves are produced inside of them. They can be made of either open or closed tubes. General Rule – Node at Closed End; Antinode at Open End Open tubes are open on both ends. Closed tubes are closed on one end.

What resonant frequency would you expect from blowing across the top of an empty soda bottle that is 18 cm deep assuming it is a closed tube? How would it change if it was 1/3 full of soda?

Beats When two frequencies of sound are produced that are similar but not identical, their interference produces beats

Doppler Effect If a sound source is stationary, the pitch heard is the same regardless of position. If a sound source is moving, the frequency changes based on the position of the observer. If the source is moving toward the observer, a higher frequency is observed (higher pitch) If the source is moving away from the observer, a lower frequency is observed (lower pitch)

The Doppler Effect is also observed with electromagnetic radiation (light). When distant stars or galaxies move at speeds that are different than that of the Earth, the light they give off changes its frequency. A galaxy that has a “RED shift” is moving away from the Earth because its waves have been stretched. A galaxy that has a “BLUE shift” is moving toward the Earth because its waves have been compressed.