Chapter 11 Vibrations and Waves

Chapter 11 is an introduction to the general subjects of oscillations. Chapter 11 discusses the effect of a steady force on an extended object. Such a force will always cause a deformation of a solid object. The deformation stores up potential energy that can be released to create oscillations and transmit wave energy.

Section 1 Simple Harmonic Motion

Hooke’s Law F = -kX

*Describe the necessary condition for which Hooke’s law is valid.

A certain spring is relaxed on a horizontal table. One end is anchored to the table. The spring is observed to stretch 3.2 cm when a force of 2 Newtons is applied to the free end. What is the value of the force constant?

 

How much will the spring stretch if a force of 4.3 Newtons is applied?

When the spring is released is the acceleration of the free end of the spring constant as the spring contracts?

*Write in definitions for displacement

amplitude

cycle

period

frequency

*What are the units used for frequency?

Many motions are periodic (spin of the earth, earth’s orbit around the sun, the motion of a second hand on a clock, etc.) Some periodic motions have additional characteristics that allow them to be called Simple Harmonic Motion ( SHM).

*In mechanical systems the motion is SHM if the magnitude of the restoring acceleration is proportional to the displacement. A spring satisfies this requirement F = -kX so

a = -(k/m)X

*Can we use the four kinematic equations to analyze SHM?

Objects on a spring undergo simple harmonic motion.

 

Section 2 Energy of a Harmonic Oscillator

A system undergoing simple harmonic motion has both potential and kinetic energy. At one point in the cycle all the energy is KE. Later, all the energy is PE.

If the motion is not damped (no energy is lost during each cycle), KE + PE = constant.

½ m V2 + ½ k A2 = Total Energy, E

E is proportional to A2

For a spring: What is the maximum velocity if k = 20 N/m, A = 5 cm and m = 3 kg?

 

 

 

Section 3 The Period and Sinusoidal Nature of SHM

The shadow of an object has simple harmonic motion if the object is moving with uniform circular motion. The study of uniform circular motion leads to the following formula for the natural period of an object moving with simple harmonic motion:

T =

The natural frequency f is f = 1/T .

If 2 kg is placed on the end of the spring discussed in section 1, what is the frequency of the motion?

 

 

For simple harmonic motion x = Vx =

ax =

 

TRUE or FALSE If for the same spring the amplitude of the motion is made larger then the period of simple harmonic motion will be greater.

 

Section 4 The Simple Pendulum

*Technically, a pendulum does not move with simple harmonic motion. The acceleration of the mass is not quite directly proportional to the arc length. If the angle of swing is less than 15 degrees then the following relationship for the period is nearly true:

T =

What is the period of a simple pendulum that has a length of 9.8 meters?

On a certain hypothetical planet an astronaut sets up a pendulum that has a length of 2 meters. She records that the pendulum has a frequency of 0.3 Hz. What is the value of the local acceleration due to gravity?

 

TRUE or FALSE A pendulum could be used to determine the altitude of an aircraft.

Section 5 Damped Harmonic Motion

What happens to the amplitude of a system that is losing energy?

What could cause this loss of energy?

Section 6 Forced Vibrations; Resonance

A forced vibration is produced by applying a force that has its own particular frequency. The frequency of the external force is the frequency of the vibration.

TRUE or FALSE It is possible for a small force to create a large amplitude for harmonic motion.

The natural frequency is f

 

Video … Tacoma Narrows Bridge

Resonance exists if the frequency of the driving force is equal to the natural frequency of the system. Name some situations in which resonance is desired and some in which it is destructive.

Section 7 Wave Motion

*A pulse is a single, localized disturbance that travels in a medium. *A periodic wave is a repeated propagation of a disturbance through a medium (except for light) without any net displacement of the medium.

*TRUE or FALSE Each particle of a water wave merely oscillates about an equilibrium point.

*Write out the definitions for wave amplitude

wavelength λ

frequency

 

*The fundamental equation of wave propagation is v = l f . It is true for all waves.

 

For waves of small amplitude on a string v =

 

Section 8 Types of Waves: Transverse and Longitudinal

For a longitudinal wave the medium oscillates in SHM parallel to the wave velocity direction. For a transverse wave the medium oscillates in SHM perpendicular to the wave velocity direction.

List two transverse waves:

List two longitudinal waves:

We will not calculate wave velocities with equations 11-14 a,b.

TRUE or FALSE A water wave is a transverse wave.

 

 

Sections 9 & 10 Energy Transported by Waves

*The energy transmitted by a wave is proportional to the square of the frequency and the square of the amplitude of the wave.

*The intensity of a wave is a measure of the amount of power transmitted through a unit of area. I = P/A e.g. watts/m2

The intensity of sunlight at the top of the earth’s atmosphere is 1,370 watts/m2 .

What minimum area of a solar collector is needed if the solar collector is to be the sole power source for a 100 watt light bulb?

What is the intensity of sunlight at the top of the mar’s atmosphere? Mars is 2.28 x 106 km from the sun.

 

Section 11 Reflection and Interference of Waves

When a pulse on a string hits a wall it reflects with a 180o phase shift.

Write down the definitions for wave front

ray

plane wave

*The principle of superposition: If two waves occupy the same space at the same time then the net displacement of the medium is the sum of the individual wave displacements. (This assumes that the medium does not exceed its elastic limit.)

If both waves have the same sign of displacement then the net displacement will be larger than each individual displacement. *This situation is called constructive interference. The two waves are said to be in phase. If the displacements have opposite signs then the net displacement will be smaller than the magnitude of the largest displacement. *This situation is called destructive interference. The two waves are said to be out of phase.

 

 

Section 12 Standing Waves; Resonance

If two interfering waves have the same wavelengths and amplitudes and opposite directions of travel a standing wave is produced. There will be locations (nodes) at which the displacement of the medium is always zero. The distance between adjacent nodes is ½ wavelength. Between the nodes are antinodes. The medium has its maximum vibration at the antinodes. There is ¼ wavelength between a node and antinode.

*The lowest frequency of a standing wave is called the fundamental frequency. For a string anchored at both ends the fundamental frequency is

f1 = V / (2L)

The harmonic frequencies are integer multiples of the fundamental frequency. fn = n * f1

 

The speed of a transverse wave on a string is given by V =

 

The material in section 13 will be covered next semester when we study optics.

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