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## Vibrations and Waves

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### Vibrations and Waves

### Simple Harmonic Motion

Chapter 11

Chapter 11 Section 1

Periodic Motion

- Any repetitive, or cyclical, types of motion
- Examples?
- Simple Harmonic Motion(SHM) is a specialized form of periodic motion

Simple Harmonic Motion

- Periodic vibration about an equilibrium position
- Restoring force must be
- proportional to displacement from equilibrium
- in the direction of equilibrium

Simple Harmonic Motion

- Common examples include:
- mass-spring system
- pendulum for small angles

Mass on a Spring

When a spring is

stretched, the restoring

force from the tension in

The spring is described

by Hooke’s Law…

F = kx

The force acting on the mass is proportional to its displacement from equilibrium and in a direction towards equilibrium, thus SHM

The Pendulum

- A simple pendulum consists of a mass called a bob, which is attached to a fixed string. Effectively, all the mass is in the bob.
- The x component of the

weight (Fg sin q) is the

restoring force.

The Pendulum

- The magnitude of the restoring force (Fgsin q) is proportional to sin q.
- When the angle of displacement q is relatively small, sin q is approximately equal to q in radians… sin 0 = 0
- So, for small angles, the restoring force is very nearly proportional to the displacement, and the pendulum’s motion is an excellent approximation ofsimple harmonic motion.

Virtual Simple Harmonic Motion

- http://phet.colorado.edu/simulations/sims.php?sim=Pendulum_Lab
- http://phet.colorado.edu/simulations/sims.php?sim=Masses_and_Springs

Measuring Simple Harmonic Motion

Chapter 11 Section 2

Amplitude

- The maximum displacement from equilibrium.

Period

- The time it takes for one complete cycle of motion.
- Represented by the symbol T
- Unit of seconds

Frequency

- The number of cycles completed in a unit of time (usually seconds)
- Represented by the symbol f
- Unit of s-1 (also known as Hertz)

Period and Frequency

- Period and frequency are inversely related.
- f = 1/T and T = 1/f

A mass-spring system vibrates exactly 10 times each second. What is its period and frequency?

f = 10 cycles per second

= 10 Hz

T = 1/f = 1/10 s

= 0.1 s

Factors Affecting Pendulums

- For small amplitudes, the period of a pendulum does not depend on the mass or amplitude.
- Length and acceleration due to gravity do affect the period of a pendulum.

Factors Affecting Mass-Spring Systems

- The heavier the mass, the longer the period (more inertia)
- The stiffer the spring, the less time it will take to complete one cycle.

Properties of Waves

Chapter 11 Section 3

What is a wave?

- A wave is an means by which energy is transferred from one place to another via periodic disturbances

Some general terminology…

- Pulse – a single disturbance, single cycle
- Periodic wave – continuous, repeated disturbances
- Sine wave – a wave whose source vibrates with simple harmonic motion
- Medium – whatever the

wave is traveling through

Mechanical Waves

- Waves that require a physical medium to travel through.
- Examples: sound, disturbance in a slinky
- Examples of physical media are water, air, string, slinky.

Electromagnetic waves

- Waves that do not require a physical medium.
- Comprised of oscillating electric and magnetic fields
- Examples include x-rays, visible light, radio waves, etc.

Transverse Waves

- Particles of the medium move perpendicular to the direction of energy transfer
- You should be able to identify crests, troughs, wavelength (distance traveled during one full cycle), and amplitude

Crest

Trough

Longitudinal Waves

- Particles of the medium move parallel to the direction of energy transfer (slinky demo)
- Be able to Identify compressions, rarefactions, wavelengths

Compressions Rarefactions

Waves transfer energy

- Note that, while energy is transferred from point A to point B, the particles in the medium do not move from A to B.
- Individual particles of the medium merely vibrate back and forth in simple harmonic motion
- The rate of energy transfer is proportional to the square of the amplitude
- When amplitude is doubled, the energy carried increases by a factor of 4.

Wave speed

- Wave speed is determined completely by the characteristics of the medium
- For an unchanging medium, wave speed is constant
- The speed of a wave can be calculated by multiplying wavelength by frequency.

v = f x λ

Practice #1

- Q: Microwaves travel at the speed of light, 3.00108 m/s. When the frequency of microwaves is 9.00 109 Hz, what is their wavelength?
- A: 0.0300 m

Practice #2

- Q: The piano string tuned to middle C vibrates with a frequency of 264 Hz. Assuming the speed of sound in air is 343 m/s, find the wavelength of the sound waves produced by the string.
- A: 1.30 m

11.3 Problems

- Page 387 1-4

Wave Interactions

Chapter 11 Section 4

5 behaviors common to all waves:

- Reflection
- Interference
- Rectilinear Propagation
- Refraction
- Diffraction

1. Reflection

- The bouncing of a wave when it encounters the boundary between two different media

Fixed End Reflection

- At a fixed boundary, waves are inverted as they are reflected.

Free End Reflection

- At a free boundary, waves are reflected on the same side of equilibrium

2. Interference

- The combination of two or more waves in a medium at the same time.
- Physical matter cannot occupy the same space at the same time, but energy can.
- The Superposition Principle describes what happens when waves interfere…
- Waves (energy) pass through each other completely unaffected
- The medium will be displaced an amount equal to the vector sum of what the waves would have done individually

Constructive Interference

- Pulses on the same side of equilibrium.
- Waves meet, combine according to the superposition principle, and pass through unchanged.
- Displacement of medium greater than originals

Destructive Interference

- pulses on opposite sides of equilibrium.
- Waves meet, combine according to the superposition principle, and pass through unchanged.
- Displacement of medium less than at least one original

Interference patterns

- Interference patterns result from continuous interference.
- http://phet.colorado.edu/en/simulation/wave-interference

Standing Waves

- An interference pattern that results when two waves of the same frequency, wavelength, and amplitude travel in opposite directions and interfere.

Standing wave parts

- Node – point that maintains zero displacement, complete destructive interference
- Antinode – point at which largest displacement occurs, constructive interference

Standing waves

- Only specific frequency-wavelength combinations will produce standing wave patterns in a given medium.

If a string is 4.0 m long, what are three wavelengths that will produce standing waves on this string?

3. Rectilinear Propagation

- Waves travel in straight lines
- The direction of travel is perpendicular to the wavefront.

Wavefront - The set of points in space reached by a wave at the same instant as the wave travels through a medium.

4. Refraction

The bending of the path of a wave as it enters a new medium of different wave speed.

5. Diffraction

- The spreading of wave energy around the edges of barriers and obstacles

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