f 1/T Formulas T 1/ f Fs kx Ts 2 m k Tp 2 l g v f
What do the following all have in common? Swing, pendulum, vibrating string
They all exhibit forms of periodic motion. Periodic Motion: When a vibration or oscillation repeats itself over the same path
Simple Harmonic Motion (SHM): A specific form of periodic motion in which the restoring force is proportional to distance from the equilibrium position.
Objects that exhibit SHM Spring Systems* Pendulums* Circular Motion Waves Sound, Light, Pressure
Definitions Period Time required for one complete cycle T =(sec/cycle) measured in seconds Frequency - Number of complete cycles in a period of time ( f =(cycle/sec) measured in Hertz Amplitude Displacement from the equilibrium position. It is a measure of energy
Definitions Equilibrium Position - The center of motion; the place at which no forces act. Displacement - The distance between the center (equilibrium position) and location of the spring, pendulum, or wave at any time.
Example 1 A fishing bobber moves up and down 24 times in 1 minute. A: What is its period? B: What is its frequency? C: What is the relationship between Period and Frequency?
Example 1 A: What is its period? T = sec/cycle T = 60 sec/ 24 bobs T = 2.5 seconds
Example 1 B: What is its frequency? f = Cycles/Sec f = 24 bobs /60 sec f =0.4 Hz
Example 1 C: What is the relationship between Period and Frequency? T = sec/cycle f = Cycles/Sec They are reciprocals of each other! T 1/ f f 1/T 1/0.4 Hz = 2.5 sec 1/2.5 seconds = 0.4 Hz
SHM and Springs Demo Vertical spring What is the natural state for the spring? What causes it to be stretched or compressed? What causes it to return to its natural state?
SHM and Springs Compare various springs How are they different? What does that mean?
Period on a Spring If we stretch a spring with a mass and release it, it will oscillate. This is SHM! period of this What is the Motion?
Horizontal Springs It has a mass of some kind attached to a spring. This spring is stretched and released. This causes the entire system to oscillate. (move back and forth)
Hooke s Law (force of a spring) FS kx F spring : magnitude of the distorting or restoring force in Newtons K: spring constant or force constant (stiffness of a spring) in Newtons per meter (N/m) x: displacement from equilibrium in meters
If time Simple harmonic motion - Physics Flash Animations
Application in Engineering and design beyond springs and rubber bands Chairs Floors Anything that flexes and provides an upward support force
FS kx Example 2 I have a slinky with a spring constant of 130 N/m. With what force do I need to pull it to stretch the slinky from its equilibrium position for the following displacements? A. 0.1m: B. 0.5 m: C. What is the relationship between Force and displacement? D. How would the required force (to displace the mass 0.1m) change if the spring constant was doubled?
FS kx Example 2 I have a slinky with a spring constant of 130 N/m. With what force do I need to pull it in order to stretch the slinky from its equilibrium position for the following displacements? A. 0.1 m Fs = (130N/m) (0.1m) = 13 N B. 0.5 m Fs = (130N/m) (0.5m) = 65 N C. What is the relationship between Force and displacement? Directly Proportional D. How would the required force (to displace the mass 0.1m) change if the spring constant was doubled? Fs = (260N/m) (0.1m) = 26 N Spring Constant is directly proportional to F
Period on a Spring The period of a spring system is given by the equation below: Ts 2 m k T the period of motion m Mass of the body attached k spring constant
Period What is the relationship between mass and period of a spring? What is the relationship between spring strength (Think spring constant) and period of a spring? Remember that period is always in seconds!
Example 3 What is the mass of my car if the shocks have a spring constant of 6000 N/m and it oscillates with a period of 2 seconds when I hit a bump in the road? Ts m kts 2 m k 4 2 2 m= (6000 N/m)(2 s) 2 /4π 2 m = 607.9 kg
What is the difference between period and frequency?
Formulas Calculating Period and Frequency T sec onds cycles T = period or time for one revolution or cycle (sec) f cycles sec ond f = number of revolutions or cycles per second (Hz or sec -1 )
Let s take a jump! http://departments.weber.edu/phys ics/amiri/director/dcrfiles/energy/b ungee4s.dcr
Refer to your definitions and answer A. The is the time of one complete vibration. B. The of vibratory motion is the number of vibrations per second. C. The frequency is the of the period.
An object suspended so that it can swing back and forth about an axis is called a. pendulum An ideal is one where all mass is considered to be concentrated in the. bob A pendulum exhibits SHM.
Foucault Pendulums demonstrates the world rotating.
https://www.youtube.com/watch?v=_ixlef9gfty Out of chaos, comes order. The scientific explanation notwithstanding, this is some neat stuff to watch Harvard built a device with a series of fifteen pendulums in a row, each one of them slightly longer than its neighbor. The pendulums were set into motion and the result was captured on video. The patterns that appear in this short video are fascinating to watch and to think about. Prepare to be captivated by this simple device! Click on the below link but before starting the video, READ the complete explanation. Fascinating. I want one! http://sciencedemonstrations.fas.harvard.edu/icb/icb.do?keyword=k16940&pageid=icb.page80863&pagecontentid=icb.pagecontent341734&state=maximize&view=view.do& viewparam_name=indepth.html#a_icb_pagecontent341734
The Pendulum Formula T = period (s) l = length (m) Tp 2 l g g = acceleration due to gravity (m/s 2 )
Refer to the pendulum formula and answer the following statements: How does mass affect period? What is the relationship between length and period? What is the relationship between acceleration of gravity and period?
Refer to the pendulum formula and answer the following statements: How does mass affect period? It doesn t! What is the relationship between length and period? period is directly proportional to the square root of its length What is the relationship between acceleration of gravity and period? period is indirectly proportional to the square root of the acceleration of gravity
Example 4: What is the period of a pendulum that is 0.35 m long at sea level? Tp 2 l g Tp 2 0.35m 9.8m / s^2 Tp 1.19sec
Example 5: The frequency of a moving pendulum measures 23 oscillations per 4.3 secs. Determine the length of the pendulum. First determine period 0.187 sec Rearrange pendulum formula to solve for length l = 0.00868 m 2 l T 4 g 2
Example 6: How do the periods of two pendulums compare if one has a measure of 25 cm and the other has a measure of 100 cm?
Example 6: How do the periods of two pendulums compare if one has a measure of 25 cm and the other has a measure of 100 cm?.25 m T = 1sec 1.0 m T = 2sec
Refer to the formula for calculation of period If one knew the length and period, what could one calculate? T 2 l g g 4 T 2 2 l GRAVITY!!!!
REVIEW: What does period of a pendulum depend on? The period of the pendulum is directly proportional to the square root of the length and inversely proportional to the square root of the acceleration due to gravity. The longer the pendulum, the greater the period.
Refer to the formula for calculation of period If one knew the length and period, what could one calculate? T 2 l g g 4 T 2 2 l GRAVITY!!!!
pendulum wave applet vibrating spring wave applet
Demo WAVES
What is a wave (continuous wave)? A repeating and periodic disturbance that transfers energy from one place to another They are an energy transport system WAVES TRANSPORT ENERGY NOT MATTER!!! The particles in a wave vibrate however they do NOT move along with the wave, only the wave front itself moves on.
What is a pulse? A pulse is a single non repeated disturbance
Types of Waves Waves are classified by 1) The use of a medium or not to carry the energy 2) The way they vibrate relative to the motion of the wave
Medium required to transfer energy Referred to as Mechanical Waves can be transmitted through solids, liquids, and gases. they can not travel through space Examples include: sound waves and water waves.
Medium NOT required to transfer energy Referred to as Electromagnetic Waves (Non Mechanical) are able to transmit energy through a vacuum as well as solids, liquids, and gases. They can travel through space: NO medium required
Examples of electromagnetic waves include cosmic, gamma, x-ray, ultraviolet, visible light, infrared, microwave, radio All waves on the EM Spectrum
ELECTROMAGNETIC WAVES All e/m waves travel through free space at a speed of approximately 3.00 x 10 8 m/s or 186,000 miles/sec. This speed is known as the speed of light c.
Categorize on direction of particle movement Longitudinal Transverse
Types of Wave Motion Longitudinal and Transverse WaveMotion Transverse Compressional (Longitudinal)
Transverse Wave Motion
Transverse Waves Motion of Molecules Direction of Wave Vibration is perpendicular (up & down) to the direction the wave is moving. ex. light waves, snakey
Transverse Wave Diagrams
Longitudinal (Compressional) Waves vibration is parallel to the direction of the wave. These waves require a medium (such as air or water) through which to travel. ex. Sound waves (looks like a spring) Direction of Movement Direction of Wave
Longitudinal Wave Motion
Cont d Compression Wavelength Rarefaction
Longitudinal Waves: Anatomy Rarefaction: region in which the particles are spread out Compression: region in which the particles are close together A wavelength: composed of a complete rarefaction and a complete compression.
Common Wave Properties Frequency and period are inversely related. T=1/f
Calculating Wave Speed: v = f Where v = wave speed in m/s f = frequency in Hz = the wavelength in meters.
Which wave has the longest wavelength?
Which wave has the greatest frequency?
What is the relationship between f and λ when velocity held constant? inversely related
IMPORTANT The speed of the wave however depends solely on the medium through which a wave is traveling
Velocity of a Wave The equation v=d/t can also be applied.
fyi The frequency of the wave is determined by the motion of the vibration of the source and the speed of a wave changes when it moves from one medium to another, therefore, the wavelength must change in response when the wave moves into a different medium.
Ex 7 A tuning fork with a frequency of 583 Hz is vibrated, generating a sound wave. Measurements indicate that the wavelength of the sound wave being generated by the tuning fork is 0.59 m long. Calculate the speed of sound in air using this information.
Ex 8 A water wave travels 94.6m in 0.285 seconds. What is the velocity of the wave? Use v = d/t 332 m/s
How can you tell How much energy a wave is going to have?
Energy and Amplitude The rate at which energy is transferred by a wave depends on the amplitude of the wave. Energy of a wave IS NOT related to the speed of the wave.
Which wave has greatest amplitude?
What is wrong here?
Let s Try: Measurements show that the wavelength of a sound wave in a certain material is 18.0 cm. The frequency of the wave is 1900 Hz. What is the speed of the sound wave? λ = 0.18 m f = 1900 Hz v = λ f = 0.18 (1900) = 342 m/s
Wave Behavior Reflection Refraction Diffraction Interference
Remember speed of a wave depends on: the medium the wave is passing through not the energy that created the vibrations. Energy only determines amplitude
What is this?
Reflection Reflection is the bouncing back of a wave at a boundary.
Law of Reflection the angle of incidence is equal to the angle of reflection Sound can also be reflected Reflected sounds are Echoes
Reflection A reflected sound wave is called an echo. The wave equation v = f as well as the equation v = d/t can both be used for sound waves.
Let s try: Assume the velocity of sound is 300m/s in a canyon. You yell hello and hear your echo 3 seconds later. How far are you from the canyon wall?
What the heck????
You tube: Amazing Water Trick http://www.youtube.com/watch?v= 8T8G_4H_TNg
What is Refraction?
Refraction Refraction is the change in speed of a wave at a boundary as it passes from one medium to another. As a result the wave bends or changes direction. The speed changes however the frequency stays the same. This means that the wavelength must change.
For refraction to occur, The wave must enter a new medium (at an oblique angle). Causing the wave to change speed & directio
Diffraction the spreading of a wave around a barrier or through an opening. The medium does not change!!!!!
These images are created by a ripple tank In order for diffraction to occur, the opening or edge must be much smaller than the incident wave
Diffraction
Diffreaction Applications Holograms (Not just depth, in it) Investigation of Molecular Structure
Double Slit Diffraction Results in constructive and destructive interference
adding waves
Interference the result of the superposition of two or more waves, i.e. two or more waves occupy the same place at the same time.
constructive vs. destructive interference Interference can be either constructive (build) or destructive (cancel). Depends on how the waves overlap
Constructive interference waves align in sync or in phase displacement is in same direction Resultant wave has greater amplitude than orignal waves.
Destructive interference waves are out of sync(out of phase) displacement is in opposite direction Resultant wave has smaller amplitude than orignal waves Total destruction if waves of equal amplitudes meet 180 O out of phase
constructive vs. destructive interference According to superposition, the displacement of the medium caused by two or more waves is the algebraic sum of the displacements caused by the individual waves. If an wave with an amplitude of +8cm has constructive interference with a wave with an amplitude of +6cm, the resulting amplitude is +14cm
Examples Interference
node vs antinode node: a point in a medium that is completely undisturbed when a wave passes. Antinode: the point of maximum displacement; it can be either a crest or a trough
Standing Wave: A result of interference Created when two periodic waves of equal amplitude and wavelength travel in the opposite direction. the nodes and antinodes of a wave are in a constant position. as the frequency of the wave increases, the number of nodes and antinodes increases in the same amount of space.