# Vibrations and Waves Chapter 12.

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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 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?

Solving the Problem Why do I make x negative?
Because the displacement is down

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

Comparison between pendulum and mass-spring system (p. 445)

Measuring Simple Harmonic Motion (p. 447)

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?

Solving the Problem

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

Solve the problem

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.

Standing Waves A N N A A N 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 N A N A N

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?

Solving the Problem What type of boundary is involved here?
Fixed So that means the pulse will be reflected and inverted What happens when two pulses meet and one is inverted? Destructive interference The resultant amplitude is 0.0 m

Helpful Simulations Mass-Spring system: Pendulum: Wave on a string system: