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## Vibrations and Waves

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**Vibrations and Waves**Chapter 12**Periodic Motion**• A repeated motion is called periodic motion • What are some examples of periodic motion? • The motion of Earth orbiting the sun • A child swinging on a swing • Pendulum of a grandfather clock**Simple Harmonic Motion**• Simple harmonic motion is a form of periodic motion • The conditions for simple harmonic motion are as follows: • The object oscillates about an equilibrium position • The motion involves a restoring force that is proportional to the displacement from equilibrium • The motion is back and forth over the same path**Earth’s Orbit**• Is the motion of the Earth orbiting the sun simple harmonic? • NO • Why not? • The Earth does not orbit about an equilibrium position**p. 438 of your book**• The spring is stretched away from the equilibrium position • Since the spring is being stretched toward the right, the spring’s restoring force pulls to the left so the acceleration is also to the left**p. 438 of your book**• When the spring is unstretched the force and acceleration are zero, but the velocity is maximum**p.438 of your book**• The spring is stretched away from the equilibrium position • Since the spring is being stretched toward the left, the spring’s restoring force pulls to the right so the acceleration is also to the right**Damping**• In the real world, friction eventually causes the mass-spring system to stop moving • This effect is called damping**Mass-Spring Demo**• http://phet.colorado.edu/simulations/sims.php?sim=Masses_and_Springs • I suggest you play around with this demo…it might be really helpful!**Hooke’s Law**• The spring force always pushes or pulls the mass back toward its original equilibrium position • Measurements show that the restoring force is directly proportional to the displacement of the mass**Hooke’s Law**• Felastic= Spring force • k is the spring constant • x is the displacement from equilibrium • The negative sign shows that the direction of F is always opposite the mass’ displacement**Flashback**• Anybody remember where we’ve seen the spring constant (k) before? • PEelastic = ½kx2 • A stretched or compressed spring has elastic potential energy!!**Spring Constant**• The value of the spring constant is a measure of the stiffness of the spring • The bigger k is, the greater force needed to stretch or compress the spring • The units of k are N/m (Newtons/meter)**Sample Problem p.441 #2**• A load of 45 N attached to a spring that is hanging vertically stretches the spring 0.14 m. What is the spring constant?**Why do I make x negative?**Because the displacement is down Solving the Problem**Follow Up Question**• What is the elastic potential energy stored in the spring when it is stretched 0.14 m?**The simple pendulum**• The simple pendulum is a mass attached to a string • The motion is simple harmonic because the restoring force is proportional to the displacement and because the mass oscillates about an equilibrium position**Simple Pendulum**• The restoring force is a component of the mass’ weight • As the displacement increases, the gravitational potential energy increases**Simple Pendulum Activity**• http://phet.colorado.edu/simulations/sims.php?sim=Pendulum_Lab • You should also play around with this activity to help your understanding**Amplitude of SHM**• Amplitude is the maximum displacement from equilibrium • The more energy the system has, the higher the amplitude will be**Period of a pendulum**• T = period • L= length of string • g= 9.81 m/s2**Period of the Pendulum**• The period of a pendulum only depends on the length of the string and the acceleration due to gravity • In other words, changing the mass of the pendulum has no effect on its period!!**Sample Problem p. 449 #2**• You are designing a pendulum clock to have a period of 1.0 s. How long should the pendulum be?**Period of a mass-spring system**• T= period • m= mass • k = spring constant**Sample Problem p. 451 #2**• When a mass of 25 g is attached to a certain spring, it makes 20 complete vibrations in 4.0 s. What is the spring constant of the spring?**What information do we have?**• M= .025 kg • The mass makes 20 complete vibrations in 4.0s • That means it makes 5 vibrations per second • So f= 5 Hz • T= 1/5 = 0.2 seconds**Day 2: Properties of Waves**• A wave is the motion of a disturbance • Waves transfer energy by transferring the motion of matter instead of transferring matter itself • A medium is the material through which a disturbance travels • What are some examples of mediums? • Water • Air**Two kinds of Waves**• Mechanical Waves require a material medium • i.e. Sound waves • Electromagnetic Waves do not require a material medium • i.e. x-rays, gamma rays, etc**Pulse Wave vs Periodic Wave**• A pulse wave is a single, non periodic disturbance • A periodic wave is produced by periodic motion • Together, single pulses form a periodic wave**Transverse Waves**• Transverse Wave: The particles move perpendicular to the wave’s motion Particles move in y direction Wave moves in X direction**Longitudinal (Compressional) Wave**• Longitudinal (Compressional) Waves: Particles move in same direction as wave motion (Like a Slinky)**Longitudinal (Compressional) Wave**Crests: Regions of High Density because The coils are compressed Troughs: Areas of Low Density because The coils are stretched**Wave Speed**• The speed of a wave is the product of its frequency times its wavelength • f is frequency (Hz) • λ (lambda) Is wavelength (m)**Sample Problem p.457 #4**• A tuning fork produces a sound with a frequency of 256 Hz and a wavelength in air of 1.35 m • a. What value does this give for the speed of sound in air? • b. What would be the wavelength of the wave produced b this tuning fork in water in which sound travels at 1500 m/s?**Part a**• Given: • f = 256 Hz • λ = 1.35 m • v = ?**Part b**• Given: • f = 256 Hz • v =1500 m/s • λ = ?**Wave Interference**• Since waves are not matter, they can occupy the same space at the same time • The combination of two overlapping waves is called superposition**The Superposition Principle**• The superposition principle: When two or more waves occupy the same space at the same time, the resultant wave is the vector sum of the individual waves**Constructive Interference (p.460)**• When two waves are traveling in the same direction, constructive interference occurs and the resultant wave is larger than the original waves**Destructive Interference**• When two waves are traveling on opposite sides of equilibrium, destructive interference occurs and the resultant wave is smaller than the two original waves**Reflection**• When the motion of a wave reaches a boundary, its motion is changed • There are two types of boundaries • Fixed Boundary • Free Boundary**Free Boundaries**• A free boundary is able to move with the wave’s motion • At a free boundary, the wave is reflected**Fixed Boundaries**• A fixed boundary does not move with the wave’s motion (pp. 462 for more explanation) • Consequently, the wave is reflected and inverted**Standing Waves**• When two waves with the same properties (amplitude, frequency, etc) travel in opposite directions and interfere, they create a standing wave.**N**N A N N N A A A N N N N A Standing Waves • Standing waves have nodes and antinodes • Nodes: The points where the two waves cancel • Antinodes: The places where the largest amplitude occurs • There is always one more node than antinode A**Sample Problem p.465 #2**• A string is rigidly attached to a post at one end. Several pulses of amplitude 0.15 m sent down the string are reflected at the post and travel back down the string without a loss of amplitude. What is the amplitude at a point on the string where the maximum displacement points of two pulses cross? What type of interference is this?