physics 1025f vibrations & waves

1UCT PHY1025F: Vibrations & Waves

Physics 1025FVibrations & Waves

Dr. Steve [email protected].

za

OSCILLATIONS


Chapter 11: Vibrations and Waves

Periodic motion occurs when an object vibrates or oscillates back and forth over the same path


Periodic motion, processes that repeat, is one of the important kinds of behaviours in Physics

Periodic Motion


Equilibrium position – position where net force is zeroRestoring force – force acting to restore equilibriumOscillation – periodic motion governed by a restoring force

Equilibrium and Oscillation


A graph or motion that has the form of a sine or cosine function is called sinusoidal. A sinusoidal oscillation is called simple harmonic motion (SHM)

Equilibrium and Oscillation


SHM is characterised by…

Amplitude A: maximum distance of object from equilibrium position

Period T: time it takes for object to complete one complete cycle of motion; e.g. from x = A to x = −A and back to x = A

Frequency ƒ: number of complete cycles or vibrations per unit time

Displacement x: is the distance measuredfrom the equilibrium point

Simple Harmonic Motion

𝑇=1𝑓


SHM occurs whenever the net force along direction of 1D motion obeys Hooke’s Law- (i.e. force proportional to displacement and always directed

towards equilibrium position)

Not all periodic motion over the same path can be classified as SHMInitially, we will look at the horizontal mass-spring system as a representative example of SHM

Simple Harmonic Motion


x is the displacement of the mass m from its equilibrium position (x = 0 at the equilibrium position)The negative sign indicates that the force is always directed opposite to displacement (i.e. restoring force towards equilibrium)

Hooke’s Law Review

k is the spring constant

spring force

sF kx


A prosthetic leg contains a spring to absorb shock as the person is walking. If an 80 kg man compresses the spring by 5 mm when standing with his full weight on the prosthetic, what is the spring constant (k)?

How far would the spring compress for a 100 kg man?

Example: Hooke’s Law


From Newton II, for a mass-spring system:

For a horizontal mass-spring system & all other cases of SHM, acceleration depends on positionSince acceleration is not constant in SHM standard “equations of motion” cannot be applied

Horizontal Mass on a Spring

netF kx maka xm


V&S Example 13.2: A 0.350-kg object attached to a spring of force constant 1.30 x 102 N/m is free to move on a frictionless horizontal surface. If the object is released from rest at x = 0.10 m, find the force on it and its acceleration at x = 0.10 m, x = 0.05 m, x = 0 m, x = -0.05 m, and x = -0.10 m.

Example: SHM


SHM occurs whenever the net force along direction of 1D motion obeys Hooke’s LawFor a pendulum, the restoring force is

Does this motion qualify as simple harmonic motion?A. YesB. No

The Simple Pendulum

sinmgFnet


A pendulum only exhibits SHM if it is restricted to small-angle oscillations (< 10°). For such small angles (in radians), we get the small-angle approximation, where

The Simple Pendulum

sin


Using the small-angle approximation, the restoring force becomes

The pendulum displacement (the arclength s) is proportional to the anglegiving

The Simple Pendulum

mgmgFnet sin

Ls

sLmg

LsmgFnet

Linear restoring force


The potential energy of a spring (Section 6-4):

The kinetic energy of the mass (Section 6-3):

Therefore the total energy of the spring-mass system is:

This total energy is conserved (assuming no friction, etc…)

Energy in a Mass-Spring System


• Energy is all PE when

• Total energy is• Energy is all KE when

• Total energy is conserved, so

Energy in Simple Harmonic Motion


A 4.0 kg mass attached to a horizontal spring with stiffness 400 N/m is executing simple harmonic motion. When the object is 0.1 m from equilibrium position it moves with 2.0 m/s. • Calculate the amplitude of the oscillation

• Calculate the maximum velocity of the oscillation

Example: Energy of Spring


Conservation of energy allows the calculation of the velocity of an object attached to a spring at any position in its motion:

Energy in Simple Harmonic Motion


• The velocity of the rotating object is equal to the maximum velocity of the object in SHM.

• The circle circumference is and the rotation time is , thus

• From energy, we have: • Combining them gives:

OR

SHM and Uniform Circular Motion


Simple Harmonic Motion• The position, velocity and acceleration are all sinusoidal• The frequency does not depend on the amplitude• The object’s motion can be written as

2 ( ) cos ty t AT

max2 ( ) sinytv t v

T

max2 ( ) cosyta t a

T


Giancoli Example 11-7: The displacement of an object is described by the following equation, where x is in meters and t is in seconds: .Determine the oscillating object’s (a) amplitude, (b) frequency, (c) period, (d) maximum speed, and (e) maximum acceleration.

Example: SHM


Using the small-angle approximation, the restoring force becomes

The pendulum displacement (the arclength s) is proportional to the angle giving

The Simple Pendulum (Review)

mgmgFnet sin

sLmg

LsmgFnet


Simple harmonic motion is based on the restoring force obeying Hooke’s Law, so let’s compare the pendulum force to Hooke’s law.

If we take , then our frequency equation becomes:

And the period equation becomes:

Frequency of Simple Pendulum

12

gfL

netmgF sL

12

kfm

netF kx

2 LTg


Two observations:– The frequency and period of oscillation

depend on physical properties of the oscillator.• Spring: Mass & Spring Constant• Pendulum: Length

– They do not depend on the amplitude of the oscillation.• Pendulum frequency does not depend on mass

The pendulum depends only on and making it a useful timing device

Frequency and Period


• Damped harmonic motion happens when energy is removed (by friction, or design) from the oscillating system.

• Resonance occurs when energy is added to an oscillator at the natural frequency of the oscillator.

Damping & Resonance


All systems have a natural frequency, the frequency at which a system will oscillate if left by itself.

Natural Frequency


Resonance occurs when energy is added to an oscillator at the natural frequency of the oscillator.

If an external force of this frequency is applied, the resulting SHM has huge amplitude!

Resonance


The basic properties of waves (the wave model) cover aspects of wave behaviour common to all waves.A wave is the motion of a disturbance.

The Wave Model

Waves carry energy & momentum without the physical transfer of material.A traveling wave is an organized disturbance with a well-defined wave speed.


Mechanical Waves… require some source of disturbance and a medium that can be disturbed with some physical connection or mechanism through which adjacent portions can influence each other (e.g. waves on a string, sound, water waves)

Two Types of Waves: Mechanical


Electromagnetic Waves... don’t require a medium and can travel in a vacuum (e.g. visible light, x-rays etc)

Two Types of Waves: Electromagnetic


A wave pulse can be created with a single ‘snap’ on a rope • Energy is transmitted from one point on the rope to the

next

A periodic (continuous) wave can be created by wiggling the rope up and down continuously• Energy is continuously being transmitted along the rope

Making a wave


Types of Mechanical Travelling WavesTransverse waves:

Longitudinal waves:

In a transverse wave, each element that is disturbed moves in a direction perpendicular to the wave motion.

In a longitudinal wave, the elements of the medium undergo displacements parallel to the motion of the wave. A longitudinal wave is also called a

compression wave.


• crests and troughs are the high and low points of a wave• amplitude, , is the height of a crest (depth of a trough)• wavelength, , is the distance between crests (troughs)• frequency, , is the number of cycles per unit time• period, , is the length of a cycle• wave velocity, , is the velocity the wave crest travels

Some definitions…

𝒗=𝝀𝑻 =𝝀 𝒇


Waves on a string (transverse waves) are propagated by the difference in directions of the tensions.Sounds waves (longitudinal waves) are pressure waves.

Waves on a String and in Air


Both waves on a string and sound waves require a medium and the properties of the medium determine the speed of the wave.For wave on a string, the speed is given by:where is the tension in the string and is the linear mass density:

Observations:- Wave speed increases with increasing tension- Wave speed decreases with increasing linear density

Wave Speed: String

sTv


Two travelling waves can meet and pass through each other without being destroyed or even altered.

The Principle of Superposition

Principle of Superposition- when two waves pass

through the same point, the displacement is the sum of the individual displacements

Pulses are unchanged after the interference.


Constructive:

Two waves, 1 and 2, have the same frequency and amplitude and are “in phase.”

The combined wave, 3, has the same frequency but a greater amplitude.

Constructive Interference


Destructive:

Two waves, 1 and 2, have the same amplitude and frequency but one is inverted relative to the other (i.e. they are 180° “out of phase”)

When they combine, the waveforms cancel.

Destructive Interference


Just like light reflects off water or an echo bounces off a cliff, a wave pulse on a string will reflect at a boundary.Whenever a traveling pulse reaches a boundary, some or all of the pulse is reflected.There are two types of boundaries:- Fixed end- Loose end

Wave Pulse Reflection


When a pulse is reflected from a fixed end, the pulse is inverted, but the shape and amplitude remains the same.

Reflection of Pulses – Fixed End

Think about Newton’s 3rd law at the boundary point.


When reflected from a free end, the pulse is not inverted, again the shape and amplitude remains the same.

Reflection of Pulses – Free End

Think about Newton’s 3rd law at the boundary point.


Pulse Refection at a DiscontinuityA discontinuity can act like a fixed or a free end depending on how the medium changes.

Low to high linear mass density acts like fixed end

High to low linear mass density acts like free end


When a travelling wave reflects back on itself, it creates travelling waves in both directions.The wave and its reflection interfere according to the Principle of Superposition.

The wave appears to stand still, producing a standing wave.

Standing Waves


A simple example of a standing wave is a wave on a string, like you will see in Vibrating String practical. The mechanical oscillator creates a traveling wave that is reflected off the fixed end and interferes with itself.The result is a series of nodes and antinodes, with the exact number depending on the oscillating frequency.

Standing Waves on a String


Nodes are points where the amplitude is 0. (destructive interference)

Anti-nodes are points where the amplitude is maximum. (constructive interference)

Distance between two successive nodes is ½ λ.



The figure shows the “n = 2” standing wave mode.

The red arrows indicate the direction of motion of the parts of the string.

All points on the string oscillate together vertically with the same frequency, but different points have different amplitudes of motion.



There are restrictions to a standing wave on a string.1. Two ends of the string are fixed,

so and must be nodes.2. Standing waves spacing is

between nodes, so the nodes must be equally spaced.

Standing Wave on a String

As a result, standing waves will only form at particular modes, which have numbers, i.e. , , etc.


Each mode has a specific wavelength.


For , the wavelength is: .

In general, the wavelength for a standing wave on a string is: for

Note: The mode number () is equal to the number of anti-nodes.


The standing wave on a string can exist only if it has one of these wavelengths: .


We can also calculate the frequency of the standing wave:

for


Standing Wave on a StringThe first mode is called the fundamental frequency: . All other modes have a frequency that are multiples of this fundamental frequency: .

The fundamental frequency () is known as the first harmonic, is the second harmonic, is the third harmonic, etc …

physics 1025f vibrations & waves

Documents

uct phy1025f

spring constant spring

spring of force constant

zerorestoring force

vibrations wavesshm

d motion

net force

restoring force equilibrium