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?