Physics Ch. 12 Vibrations and Physics Ch. 12 Vibrations and WavesWaves

Any repeated motion over the same path is called a periodic motion

Ex: a child swinging on a swing, a pendulum on a grandfather clock, an acrobat swinging on a trapeze

Fig. 12-1 mass attached to a springThe spring exerts a force on the mass when

the spring is stretch or compressed.

At unstretched position, the spring is at equilibrium.

The force decreases as the mass approaches equilibrium and becomes zero at equilibrium

The mass’s acceleration becomes zero at equilibrium

The velocity reaches maximum at equilibrium. The mass’s momentum causes the mass to

overshoot the equilibrium and compress the spring

12-1c: When the spring’s compression is equal to the stretched distance, the mass is at maximum displacement and the spring force and acceleration reach their maximum

Velocity at this point reaches zero.The force works in opposite direction of the

mass’s direction.

In an ideal situation, the mass would vibrate back and forth indefinitely

In the real world, friction reduces the motion and it eventually stops.

This effect is called damping.

Restoring force-the spring force pushing the mass back toward its original equilibrium position.

Restoring force directly proportional to the displacement of the mass. (Simple Harmonic Motion)

Hooke’s LawHooke’s Law

Felastic = -Kx

Spring force = -(spring constant X displacement)

K = spring constant (measure of the stiffness of the spring)

Sample Problem 12ASample Problem 12A

If a mass of 0.55 kg attached to a vertical spring stretches the spring 2.0 cm from its original equilibrium position, as shown in fig. 12-2, what is the spring constant?

Practice ProblemPractice Problem

A 76 N crate is attached to a spring (k=450 N/m). How much displacement is caused by the weight of this crate?

Practice ProblemPractice Problem

A spring of k=1962 N/m loses its elasticity if stretched more than 50.0 cm. What is the mass of the heaviest object the spring can support without being damaged?

Elastic Potential EnergyElastic Potential Energy

A stretched or compressed spring has stored elastic potential energy.

Ex: an archer that pulls the bowstring back has only elastic potential energy

When the bowstring is released, the elastic potential energy is converted to kinetic energy.

Because energy is conserved, the elastic potential energy is converted to kinetic energy of the arrow, bow, and bowstring.

The Simple PendulumThe Simple Pendulum

A simple pendulum has a mass, called a bob, attached to a fixed string.

We assume that the mass of the bob is concentrated at a point and the mass of the string is negligible. We also disregard the effects of friction or air resistance.

For a physical pendulum, such as an acrobat, we will assume the same.

Restoring ForceRestoring Force

Which force on a pendulum acts as the restoring force?

The forces on the pendulum are the force of the string and the force of the bob’s weight. The force of the weight can be split into two components, x and y. The y component is opposite the force of the string and cancels. The net force on the bob is the x component of its weight.

This force then pushes or pulls the motion toward equilibrium, the restoring force.

The restoring force varies with the position to the equilibrium

Decreases as it approaches equilibrium and becomes zero at equilibrium

At small angles (<15°) the motion is simple harmonic

At maximum displacement, the restoring force and the acceleration are at maximum while the velocity is zero.

At equilibrium, the restoring force and acceleration become zero and velocity reaches a maximum

Table 12-1

Conservation of EnergyConservation of Energy

Spring-elastic potential energyPendulum-gravitational potential energyAt equilibrium, the gravitational potential

energy is at zero, while the energy is solely kinetic energy

At maximum displacement, the kinetic energy turns entirely to gravitational potential energy

QuickLab pg. 444QuickLab pg. 444

Energy of a Pendulum

12-2: Measuring Simple 12-2: Measuring Simple Harmonic MotionHarmonic Motion

Amplitude-maximum displacement from equilibrium (Can be measured by the angle or by the amount the spring is stretched or compressed)

Period (T)-the time it takes a complete cycle of motion; the pendulum ends of where it started

Frequency (f)-the number of cycles per second or the SI unit is Hertz (Hz)

These two units are inversely related:

F = 1 or T = 1

T f

Very Quick Lab-4 Period and Very Quick Lab-4 Period and FrequencyFrequency

Attach the pendulum bob to the string and suspend the string from the ring stand. Set the pendulum in motion. Have a student record the time required to complete 20 oscillations. Then, have another student record how many times the pendulum bob returns to the same place each second.

Have students use the first measurement to find the pendulum’s period. Use T=number of seconds/20 Ask the students what the second measurement indicates. Compare the values.

Period of a pendulum depends on Period of a pendulum depends on

Length of pendulumLength of pendulum If you have two pendulums with two

different small amplitudes, the periods would still be the same. Thus, the period does not depend on the amplitude (<15º).

However, if the length of the pendulum string were changed or the free-fall acceleration, then the period would change.

Period of a simple pendulum Period of a simple pendulum in simple harmonic motionin simple harmonic motion

T = 2πL

g

Period=2π X square root of (length/free-fall acceleration)

Did you know?Did you know?

Galileo is credited as the first person to notice that the motion of a pendulum depends on its length and is independent of its amplitude (for small angles). He supposedly observed this while attending church services at a cathedral in Pisa. The pendulum he studied was a swinging chandelier that was set in motion when someone bumped it while lighting the candles. Galileo is said to have measured its frequency, and hence its period, by timing the swings with his pulse.

Very Quick Lab-5-Relationship between Very Quick Lab-5-Relationship between the length and the period of a pendulumthe length and the period of a pendulum

Repeat Lab 4 with a variety of lengths. Record each length and its corresponding period. (Frequency does not need to be measured here) Verify that the results are consistent with the following equation:

T = 2πL gCalculate the length required to have a period of 1.0

s. Construct the pendulum to test to your prediction.

Why does length affect Why does length affect period?period?

When two strings have the same angle but different lengths, the arcs that the bob must travel to equilibrium is different and therefore, the periods will be different.

Fig. 12-9

Why don’t mass and Why don’t mass and amplitude affect period?amplitude affect period?

A heavier mass will provide a large restoring force, but it also needs a larger force to achieve the same acceleration.

Because the acceleration is the same, the periods will be the same.

Similarly, the larger amplitude requires a larger restoring force. The acceleration will be greater but the distance to move is also greater. The period would stay the same.

Sample Problem 12BSample Problem 12B

Practice ProblemsPractice Problems

What is the period of a 3.98 m long pendulum? What is the period of a 99.4 cm long pendulum?

Periods of a mass-spring Periods of a mass-spring systemsystem

Period of a mass-spring system depends on mass and spring constant.

A heavier mass attached to a spring increases inertia without providing a compensating increase in restoring force

A heavy mass has a smaller acceleration than a light mass has. So, a heavy mass has a greater period.

As mass increases, so does the period.

The greater the spring constant (k), the stiffer the spring.

A stiffer spring will take less time to complete one cycle of motion than one that is less stiff.

As with the pendulum, changing the amplitude of the vibration does not affect the period.

Period of a mass-spring system in Period of a mass-spring system in simple harmonic motionsimple harmonic motion

T = 2πm

k

Period=2π X square root of (mass/spring constant)

Sample Problem 12CSample Problem 12C

Practice Problem 12CPractice Problem 12C

A 1.0 kg mass attached to one end of a spring completes one oscillation every 2.0 s. Find the spring constant.

Conceptual ChallengeConceptual Challenge

Why is a pendulum a reliable time-keeping device, even if its oscillations gradually decrease in amplitude over time?

12-3: Wave Motion12-3: Wave Motion

A wave motion travels away from the disturbance that causes the wave.

Particles in the medium vibrate up and down as the wave passes.

Ex: a leaf in a pond wave Almost all wave types need a medium to travel

through, these are called mechanical waves. Electromagnetic waves do not need a medium and

can travel in outer space Ex: x-rays, microwaves, etc.

Wave TypesWave Types

Pulse waves-a single traveling pulseEx: Fig. 12-11; a single flip of the wrist

while holding one end of a rope that is fixedPeriodic wave-continuous pulses that form

periodic motion such as moving your hand up and down repeatedly while holding a rope

Fig. 12-12As the sine wave created by this vibrating

blade travels to the right, a single point on the string vibrates up and down with simple harmonic motion

Transverse Waves-particles of the medium vibrate perpendicularly to the motion of the wave

Fig. 12-13 waveform-represents the displacements at a moment in time or the displacements of a single particle as time passes

Crest-the highest point above equilibriumTrough-lowest point below equilibriumAmplitude-maximum displacement from

equilibriumFig. 12-13bA wave is a cyclical motion, first displaced

in one direction, then in the other direction, then returning to equilibrium.

Wavelength-the distance the wave travels during one cycle, λ

Crest to crest=wavelengthTrough to trough=wavelength

If a spring is pumped back and forth toward the opposite fixed end, a longitudinal wave is formed

Vibrations are parallel to the motion of the wave The spring should have compressed and stretched

regions of the coil that travel along the spring Ex: sound waves Fig. 12-15 the compressed regions correspond to

the crests and the stretched regions correspond to troughs

Compressed regions are high density and high pressure

Stretched regions are low density and pressure

Frequency, Period & Wave Frequency, Period & Wave SpeedSpeed

Frequency of the vibrations of the source of a wave equal the vibrations of the particles in the wave

Wave frequency describes the number of crests or troughs that pass a given point in a unit of time

Period of a wave describes the time it takes for a complete wavelength to pass a given point

Period is inversely related to frequency

A displacement of one wavelength occurs in a time interval equal to one period of the vibration

V=λ/T Substitute the inverse relationship, f=1/T

Into this equation gives us

V=fλ where the speed of a wave is constant except when traveling from one medium to another

Sample Problem 12DSample Problem 12D

Waves carry energy as they move across a medium while the medium doesn’t move

Waves transfer energy by transferring motion of matter rather than by transferring matter itself, they do so efficiently

The greater the amplitude, the more energy carried in a given time

The energy transferred is proportional to the square of the wave’s amplitude

Ex: if amplitude of a wave is doubled, the energy is increased by a factor of four

If the amplitude is halved, the energy is decreased by a factor of four

12-4: Wave Interactions12-4: Wave Interactions

Two mechanical waves can occupy the same space at the same time because they are only displacements of matter, not matter

When two waves pass through one another, they form interference patterns of light and dark bands called superposition, fig. 12-17

Ex: electromagnetic radiation waves may also interfere

Demo 10Demo 10

Wave superposition

Constructive InterferenceConstructive Interference

When two waves are traveling toward each other with the same direction displacements, they form a resultant wave that is equal to the sum of the individual waves, called superposition principle.

If the waves are on the same side of the equilibrium, it is called constructive interference.

Once the waves pass through each other, their individual displacements are equal to their previous displacements

Fig. 12-18

Destructive InterferenceDestructive Interference

When the displacements are in opposite directions from the equilibrium, the resultant wave is the difference between the pulses, called destructive interference

Fig. 12-19

Complete Destructive Complete Destructive InterferenceInterference

When two pulses have equal but opposite amplitudes, the resultant wave has a displacement of zero, called complete destructive interference.

Fig. 12-20After the interference, each pulse resumes

its previous amplitude

The superposition principle is true for longitudinal waves as well

A compression is a force on a particle in on direction, while a rarefaction involves a force on the same particle in the opposite direction.

ReflectionReflection

At a free boundary, waves are reflected due to an upward force at the boundary.

Fig. 12-21 a

At a fixed boundary, waves are reflected and inverted due to a downward force at the boundary.

Fig. 12-21b

Standing WavesStanding Waves

When a string is shaken up and down in a regular motion, it produces waves with the same frequency, wavelength, and amplitude. Those waves are then sent down the string and reflected back down the string toward each other.

The result is a standing wave-a wave pattern that does not move down the string.

A standing wave contains both constructive and destructive interference

Fig. 12-22aNodes-points at which two waves cancel,

complete destructive interferenceAntinodes-point at which the largest

amplitude occurs, constructive interference

Fig. 12-23Fig. 12-23

Waves running together create a blur pattern

A single loop represents a crest or trough alone while two loops correspond to a crest and a trough together, one wavelength

The ends of the strings must be nodes, thus it is a standing wave

Fig. 12-23 Standing Fig. 12-23 Standing Waves/WavelengthWaves/Wavelength

The wavelengths of these standing waves depends on the string lengths

Wavelengths:A. String length, LB. 2LC. LD. 2/3L

Demo 10-Wave SuperpositionDemo 10-Wave Superposition

Two students hold long coiled springOne student send a single pulse down the

springThe opposite student then sends an identical

pulse.Both students generate pulses

simultaneously.

Where do the pulses cross each other? How can we tell?How can we tell that the pulses are crossing

through each other and not bouncing off each other?

Now, send two pulses of very different amplitudes toward each other.

Demo 11-Waves passing each Demo 11-Waves passing each otherother

Same scenario as demo 10Create pulses with opposite displacements

and observe the waves that reach the hands of the students

Observe the same using waves with different amplitudes and displacements on the same side.

Observe constructive and destructive

Demo 12-Wave ReflectionDemo 12-Wave Reflection

Fix one each of a spring to an object and hold the other in your hand.

Send a pulse down the spring and observe the reflected pulse.

Is the pulse inverted?Try the same with a piece of rope fixed and

then loosely tied to an object.

Demo 13-Standing WavesDemo 13-Standing Waves

Create a standing wave with the springs