IB Physics Unit 4 Oscillations Notes

7.1 | Waves – Sound

IB Physics Content Guide

Big Ideas

  •         Simple harmonic motion is a repeating relationship between an object’s position, velocity, and acceleration
  •         Waves are formed and transferred by particles oscillating in a medium
  •         All waves have properties can be measured and mathematically related
  •         Instruments resonate at specific frequencies due to the number of standing waves that fit in the length of the system
  •         Waves can occupy the same space at the same space to create constructive or destructive interference

Content Objectives

Simple Harmonic Motion

  • I can qualitatively describe the motion of an oscillating system
  • I can relate the acceleration of an object in simple harmonic motion to its position
  • I can graph the displacement, velocity, and acceleration vs time for simple harmonic motion
  • I can interpret an SHM graph to describe the conditions at a specific point in an object’s motion
  • I can describe and relate the properties of period and frequency
  • I can calculate period and frequency from a scenario
  • I can qualitatively describe the energy changes that take place during an oscillation

Properties of Traveling Waves

  • I can describe how waves carry energy through a medium
  • I can compare the properties of transverse and longitudinal waves
  • I can identify a wave example as transverse or longitudinal
  • I can read a wave’s amplitude, wavelength, period, and frequency from a graph
  • I can label a graph with the location of a wave’s crest/compression and trough/rarefaction
  • I can describe the number of complete wavelengths represented in a picture
  • I can use the wave speed equation to mathematically relate speed, wavelength, and frequency
  • I can relate pitch and frequency for sound waves

Standing Waves and Sound

  • I can describe the motion of a standing wave
  • I can identify and label the node and antinodes on a standing wave diagram
  • I can calculate the wavelength of a standing wave for different harmonics
  • I can describe how harmonics make it possible for one system to resonate at different frequencies
  • I can describe the end conditions and nodes/antinodes for open/closed pipes and vibrating strings
  • I can relate length and wavelength for open/closed pipes and vibrating strings
  • I can calculate the length of a pipe/string required to resonate a specific frequency

Speed of Sound and Wave Interference

  • I can describe why sound travels at different speeds in different media
  • I can calculate how far a distant object is by timing an echo
  • I can qualitatively and quantitatively interpret cases of constructive and destructive interference
  • I can add up two waves with superposition to create a new waveform
  • I can describe applications and real-world examples for wave interference
  • I can use wavelength and source distance to identify maxima and minima for interference

7.1 | Waves – Sound

Shelving Guide

 

Variable Symbol

Unit

 

Data Booklet Equations:

Period

T

s

 

Frequency

f

Hz

 

Wavelength

λ

m

 

Amplitude

A

m

 

 

Wave Speed

v

m s-1

 

 

Simple Harmonic Motion Graphs

Velocity vs Displacement

Accel. vs Displacement

Types of Waves

Picture

Definition

Examples

Transverse

https://sites.google.com/site/waveslightandsoundunit/_/rsrc/1467886271961/03---unit-lessons/03---sound-waves/Wave%20-%20Sound%20wave.gif?height=204&width=357

Particles move perpendicular to the motion of the wave

·     Light

·     Ripples in a Pond

·     Earthquakes

Longitudinal

https://sites.google.com/site/waveslightandsoundunit/_/rsrc/1467886271961/03---unit-lessons/03---sound-waves/Wave%20-%20Sound%20wave.gif?height=204&width=357

Particles move parallel

to the motion of the wave

·     Sound

·     Earthquakes

     

Parts of a Wave

Label the Wave:

·        Amplitude

·        Wavelength

·        Crest

·        Trough

Harmonics

 

Open Pipe

Closed Pipe

String

End Conditions

Antinode

Antinode

Node

Antinode

Node

Node

3rd Harmonic

2nd Harmonic

1st Harmonic

(Fundamental)

Interference

Constructive

Path Difference = n λ

Destructive

Path Difference = (n + ½) λ

Oscillations: “Any motion in which the displacement of a particle from a fixed point keeps changing direction and there is a periodicity in the motion, i.e. the motion repeats in some way.” (Tsokos, 2014)

​Simple Harmonic Motion (SHM)​

  • Definition: “Motion in which (the magnitude of) acceleration is proportional and opposite to displacement from a fixed (equilibrium) position (where x = 0).”​​
  • Constant quantities: Amplitude, period and frequency. Definitions below for SHM:
    • Amplitude (A or xo): Maximum displacement from equilibrium position.
    • Period (T): ​Time take
    • n to complete one full oscillation. Unit: s.
    • Frequency (f): Number of oscillations completed in one second. Unit: hertz (Hz)

Examples:​

  • Simple pendulum:​​

pendulum.gif

Mass-spring system:

Spring.gif

Graphical representation (SHM)​

  • Acceleration-displacement: Negative gradient and direct proportionality.
    • Maximum acceleration at amplitude, zero acceleration at equilibrium position.​
  • Energy-displacement:
    • Total energy (always constant) = Kinetic energy (EK)​ + Potential energy (PE).
  • Displacement, velocity and acceleration versus time:
    • Sine or cosine functions of time.
    • Phase difference (shift) between graphs:
      • Displacement-time and velocity-time: 0.25T​.
      • Displacement-time and velocity-time: 0.50T.
      • Velocity-time and acceleration-time: 0.25T.
      • Think in terms of Calculus!
        • Acceleration as the derivative of velocity.
        • Velocity as the derivative of velocity.
    • When the phase difference is zero or T, the systems are in phase.​

PERIODIC AND OSCILLATORY MOTION

PERIODIC MOTION

When a body repeats its motion after regular interval of time, it is said to be in periodic motion. The path of periodic motion may be rectilinear, open/closed curvilinear. Example:

  • Motion of moon around earth
  • Motion of a piston in a cylinder
  • Motion of a simple pendulum etc.

OSCILLATORY MOTION

If during a periodic motion, the particle moves to and fro on the same path, the motion is vibratory or oscillatory.Example :

  • The motion of a ball in bowl
  • The needle of a sewing machine
  • Vibrations of prongs of tuning fork etc.

Note:-

  • All oscillatory motion are periodic but all periodic motion are not oscillatory motion.
  • The oscillatory motion which can be expressed in terms of sine and cosine function, is said to be harmonic motion.

SIMPLE HARMONIC MOTION (S.H.M.)

If a particle moves up and down (back and forth) about a mean position (also called equilibrium position) in such a way that a restoring force/ torque acts on particle, which is proportional to displacement from mean position, but in opposite direction from displacement, then motion of the particle is called simple harmonic motion. If displacement is linear, it is called linear S.H.M. and if displacement is angular, it is called an angular S.H.M.Example :

  • Motion of a body suspended by a spring
  • Oscillations of simple pendulum

EQUATIONS OF S.H.M.

LINEAR S.H.M.The restoring force is proportional to the displacement from mean position.i.e., F ∝ – xor, where k is called force constant or spring constantor,

ANGULAR S.H.M.The restoring torque is proportional to the angular displacement from the mean position. where C is called torsional rigidityor, or, 

TERMS RELATED TO S.H.M.

  • Amplitude : The maximum displacement of the oscillating particle on either side of its mean position is called its amplitude. It is denoted by A.
  • Time period : The time taken by a oscillating particle to complete one oscillation is called its time period. It is denoted by T.
  • Frequency : It is the number of oscillations completed in one second.
    It is denoted by υ.

The S.I. unit of frequency is s–1 or Hz.

  • Angular frequency

The S.I. unit of angular frequency is rad/sec.

  • Phase : The parameter, by which the position of particle from its mean position is represented, is known as phase. 
  • The phase at any instant tells the state of position & direction of motion at that instant. The phase at time t = 0 is known as the initial phase or epoch (e).
  • Total phase angle : The total angle (ωt + θ) is known as total phase angle.

CHARACTERISTICS OF S.H.M.

DISPLACEMENT
The displacement of a particle in S.H.M. is given bywhere A is amplitude, ω is angular frequency and (ωt + φ) is called the phase of the particle at any instant t.

VELOCITY
The velocity of a particle in S.H.M. is given by or, At y = 0 (at mean position),

ACCELERATION : The acceleration of a particle in S.H.M. is given byor a = – ω2yThe negative sign indicates that the acceleration is directed towards the mean positionAt y = A (at extreme position),  amax = – ω2A

ENERGY

  • Kinetic energy : A particle in S.H.M. possesses kinetic energy by virtue of its motion.

  • Potential energy : A particle in S.H.M. possesses potential energy due to its displacement from the mean position.

  • Total mechanical energy

E = K.E. + P.E.

The curves representing KE, PE and total energy are shown in figure.

KEEP IN MEMORY

  1. Restoring force F = – Mω2x
  2. Kinetic energy = (1/2) Mω2(A2 – x2)
  3. Potential energy = 1/2 Mω2×2
  4. Total energy of SHM = 1/2 Mω2A2

Equation a = – ω2y shows that if body perform S.H.M. then acceleration of the body is proportional to displacement, but in the opposite direction of displacement. It is an essential requirement for any motion to be S.H.M.

  1. The kinetic and potential energy of SHM varies sinusoidally with a frequency twice that of SHM.
  2. Total energy

where n = frequency of vibration.

  1. where ω is constant

  1. Geometrically the projection of the body undergoing uniform circular motion on the diameter of the circle is SHM.
  2. In a non-inertial frame.

SOME SYSTEMS EXECUTING S.H.M.

CASE 1 –  Spring mass system

  • When two springs having force constants k1 and k2 connected in parallel, then

The force constant of the combination is k = k1 + k2 and hence T = 2π[M/(k1 + k2)]1/2

  • When two springs of force constants k1 and k2 are connected in series, then

The force constant of the combination is 1/k = 1/k1 + 1/k2. i.e., k = k1k2/(k1 + k2) Hence

  • If two mass M1 and M2 are connected at the two ends of the spring, then their period of oscillation is given by

T = 2π[μ/k)]1/2 where is the reduced mass.

  • When the length of spring increases, spring constant decreases. If the length of spring becomes n times, its spring constant becomes times and therefore time period will be increased by times.
  • If we divide the spring into n equal parts, the spring constant of each part becomes n k. Hence time period when the same mass is suspended from each part is:

CASE 2 – Simple pendulumA simple pendulum consists of a point mass suspended by a weightless inextensible cord from a rigid support.

Let a bob of mass m is displaced from its, equilibrium position and released, then it oscillates in a vertical plane under gravity. Let θ be the angular displacement at any time t, then corresponding linear displacement along the arc isx = l θ.It is clear from the diagram that mg sinθ, is the restoring force acting on m tending to return it to mean position. So from Newton’s second law   …(i)where negative sign indicates that restoring force mg sin θ (= F) is opposite to displacement θ. If θ is very small, then
sin θ ≈ θ, so from equation (i) …(ii) where ω2 = g/l.This is the equation of S.H.M. of the bob with time period

How to find the time period of a body undergoing S.H.M.?Step 1 : First, find the equilibrium position. Equilibrium position will be one for which and Step 2 : Displace the body from the equilibrium position by x.Find the restoring force acting on the body F = –kx (for translation)Find the restoring torque acting on the body (for rotational)Step 3 : Since ∴ Use  … (i) for translational       … (ii) for rotationalStep 4 : (for translational)   (for rotational) where, I = moment of inertia

Common DefaultIncorrect. The time period of spring mass-system is dependent on the value of g. Correct. Time period of spring-mass system shifts only the equilibrium position. It does not change the time period. Because of this reason, time period of spring mass system remains same on plains / mountains / in satellites.Incorrect. The time taken to cover half the amplitude form equilibrium position is .Correct. The actual time taken is .Incorrect. In a spring mass system, mass oscillate about the end of a spring when the spring is in unstretched condition.Correct. The mass oscillates about the equilibrium position which may or may not be at the unstretched length.

CASE 3 – Liquid in U-tubeA U-tube of uniform cross-sectional area A has been set up vertically with open ends facing up.

[Restoring force = –2Adg x] If m gm of a liquid of density d is poured into it then time period of oscillation.

CASE 4 – Rectangular block in liquidRectangular block floating in a liquid,
 
where d = density of liquid, d′ = density of block, h = height of block

CASE 5 – Vibration of gas system in a cylinder with frictionless piston.Time period, where m = mass of gas, A = cross sectional area of pistonP = pressure exerted by gas on the piston, h = height of piston

CASE 6 – If a tunnel is dig in the earth diametrically or along a chord then time period, for a particle released in the tunnel.CASE 7 – The time period of a ball oscillating in the neck of a chamber

CASE 8 – If a dipole of dipole moment p is suspended in a uniform electric field E then time period of oscillation  KEEP IN MEMORY

  1. In S.H.M. the phase relationship between displacement, velocity and acceleration, is as follows :
    1. The velocity is leading the displacement by a phase radian
    2. The acceleration is leading the displacement by a phase π radian
    3. The acceleration is leading the velocity by a phase radian.
    4. When , then velocity V = 0.86Vmax.
    5. When V = Vmax/2, the displacement x = 0.87A.
    6. When , the kinetic energy of S.H.M. is 75% of the total energy and potential energy 25% of the total energy.
    7. When the kinetic energy of S.H.M. is 50% of the total energy, the displacement is 71% of the amplitude .
  2. The time period of a simple pendulum of length l which is comparable with radius of earth.

where R = radius of the earth and g is the acceleration due to gravity on the surface of the earth.

    1. When l << R, then
    2. When l = R, we find
    3. When l = , then

= 84.6 minutes. Thus maximum of T is 84.6 minutes.

    1. Under weightlessness or in the freely falling lift

      This means, the pendulum does not oscillate at all.
  1. Time period of a simple pendulum in a train accelerating or retarding at the rate a is given by

If a simple pendulum whose bob is of density do is made to oscillate in a liquid of density d, then its time period of vibration in liquid will increase and is given by

(where d0 > d)

The time period of a simple pendulum in a vehicle moving along a circular path of radius r and with constant velocity V is given by,

If T1 and T2 are the time periods of a body oscillating under the restoring force F1 and F2 then the time period of the body under the influence of the resultant force will be

  1. (a) The percentage change in time period of simple pendulum when its length changes is

(b) The percentage change in time period of simple pendulum when g changes but l remains constant is (c) The percentage change in time period of simple pendulum when both l and g change is

  1. If a wire of length l, area of cross-section A, Young’s modulus Y is stretched by suspending a mass m, then the mass can oscillate with time period
  2. If a simple pendulum is suspended from the roof of compartment of a train moving down an inclined plane of inclination θ, then the time period of oscillations

  1. If a ball of radius r oscillates in a bowl of radius R, then its time period is given by :
  2. If a disc of radius r oscillates about a point at its rim, then its time period is given by:

It behaves as a simple pendulum of length r.

  1. The graph between the length of a simple pendulum and its time period is a parabola.
  2. The graph between the length of a simple pendulum and the square of its time period is a straight line.
  3. The graph between l & T and between l & T2 intersect at T = 1 second.
  4. The time period of the mass attached to spring does not change with the change in acceleration due to gravity.
  5. If the mass m attached to a spring oscillates in a non-viscous liquid density σ, then its time period is given by

where k = force constant and is density of the mass suspended from the spring.

  1. The length of second pendulum (T = 2 sec) is 99 cm

PHYSICAL PENDULUM

Trestoring = – mgd sin θ If θ is small, sin θ ≈ θ∴ Trestoring = – mgdθ and Let a test-tube of radius r, carrying lead shots of mass m is held vertically when partly immersed in liquid of density ρ. On pushing the tube little into liquid and let it executes S.H.M. of time period

CONICAL PENDULUM

When the bob of a simple pendulum moves in a horizontal circle it is called as conical pendulum.

If l is the length of the pendulum and the string makes an angle θ with vertical then time period,

TORSIONAL PENDULUM

It is an arrangement which consists of a heavy mass suspended from a long thin wire whose other and is clamped to a rigid support. Time periodwhere  I = moment of inertia of body about the suspension wire as axis of rotation.C = restoring couple per unit thirst.

KEEP IN MEMORY

  • The displacement, velocity and acceleration of S.H.M. vary simple harmonically with the same time period and frequency.
  • The kinetic energy and potential energy vary periodically but not simple harmonically. The time period of kinetic energy or potential energy is half that of displacement, velocity and acceleration.
  • The graph between displacement, velocity or acceleration and t is a sine curve. But the graph between P.E. or K.E. of S.H.M. and time t is parabola.
  1. If the bob of simple pendulum is -vely charged and a +vely charged plate is placed below it, then the effective acceleration on bob increases and consequently time period decreases.

Time period, In this case electric force q E and gravity force act in same direction.

  • If the bob of a simple pendulum is -vely charged and is made to oscillate above the -vely charged plate,

then the effective acceleration on bob decreases and the time period increases.

In this case electric force qE and gravity force are opposite.

  1. A pendulum clock slows down in summer and goes faster in winter.
  2. Potential energy of a particle executing S.H.M. is equal to average force × displacement.

i.e., .

  1. If the total energy of a particle executing S.H.M. is E, then its potential energy at displacement x is

and kinetic energy

FREE, DAMPED, FORCED OSCILLATIONS AND RESONANCE

FREE OSCILLATION

If a system oscillates on its own and without any external influence then it is called as free oscillation. Frequency of free oscillation is called natural frequency. The equation for free S.H.M. oscillation= Frestoring force = –kx, where k is constant.The differential equation of harmonic motion in absence of damping and external force is , where ω0 is natural frequency of body. The time period is

DAMPED OSCILLATION

Oscillation performed under the influence of frictional force is called as damped oscillation.In case of damped oscillations the amplitude goes on decreasing and ultimately the system comes to a rest.The damping force (Fdamping ∝ – v ⇒ Fdamping = – bv) is proportional to the speed of particle. Hence the equation of motion where b is positive constant and is called damping coefficient. Then the differential equation of a damped harmonic oscillation is  …(i)where 2 C = b/m (C is damping constant) & , the natural frequency of oscillating particle i.e., its frequency in absence of damping.

In case of overdamping the displacement of the particle is sin (ωt + φ)  …(ii)⇒ x = A sin (ωt + φ) where A0 = max. amplitude of the oscillator. and (relaxation time)It is clear from the fig & eqn. (ii) that the amplitude of damped harmonic oscillator decreases with time. In this case, the motion does not repeat itself & is not periodic in usual sense of term. However it has still a time period, , which is the time interval between its successive passage in same direction passing the equilibrium point.

FORCED OSCILLATION AND RESONANCE

The oscillation of a system under the action of external periodic force is called forced oscillation. External force can maintain the amplitude of damped oscillation. When the frequency of the external periodic force is equal to the natural frequency of the system, resonance takes place.

The amplitude of resonant oscillations is very very large. In the absence of damping, it may tend to infinity. At resonance, the oscillating system continuously absorbs energy from the agent applying external periodic force. In case of forced oscillations, the total force acting on the system is  …… (i)Then by Newton’s second law : ⇒ or   …… (ii)where The equation (ii) is the differential equation of motion of forced harmonic oscillator. The amplitude at any time t iswhere & p is the frequency of external periodic force.

4.1.1 Describe examples of oscillations.

  • A pendulum

  • A mass attached to a spring attached to a wall that oscillates back and forth.

4.1.2 Define the terms displacement, amplitude, frequency, period and phase difference.

Displacement – The instantaneous distance of the moving object from its mean position

Amplitude  The maximum displacement achievable from the mean position

Frequency – The number of oscillations completed per unit time

F = 1/t

Period – the time taken for a complete oscillation

T = 1/f

Phase difference – the measure of how “in step” different particles are. If they are moving together they are said to be in phase. If not they are said to be out of phase.

4.1.3 Define simple harmonic motion (SHM) and state the defining equation as a=-ω2x.

Simple harmonic motion is defined as the motion that takes place when the acceleration, a , is always directed towards and is proportional to its displacement from a fixed point.

The acceleration is caused by a restoring force that always pointing to the mean position and is proportional to the displacement from the mean position.

a = −ω²x

The negative signs signifies that the acceleration is always is always pointing back towards the mean position

4.1.4 Solve problems using the defining equation for SHM.

4.1.5 Apply the equations x = sin(ωt) , v = Aωcos(ωt), a = -Aω2 sin(ωt), v =±ω√((A2-x2 ) ) as solutions to the defining equation for SHM.

Elastic potential energy EEP = 1/2 kx2

Energy changes during simple harmonic motion (SHM)

4.2.1 Describe the interchange between kinetic energy and potential energy during SHM.

In a SHM motion the total energy is interchanged between kinetic energy and potential energy. If no resistive forced acts on the motion the total energy is constant and is said to be undamped.

Potential energy increases as we move away from the equilibrium position and kinetic energy decreases. As we come closer to the equilibrium position its vice versa. EP can be expressed as a sine curve, EK as a cosine curve.

4.2.2 Apply the expressions EK=½mω2(x02-x2) for kinetic energy of a particle undergoing SHM, ET=½mω2×02 for the total energy and EP=½mω2×2 for the potential energy.

Forced oscillations and resonance

4.3.1 State what is meant by damping.

Damping is a dissipating force that is always in the opposite direction to the direction of motion of the oscillating particle. As work is being done against the dissipating force energy is lost. Since energy is proportional to the amplitude, the amplitude decreases exponentially with time.

4.3.2 Describe examples of damped oscillations.

Under damped -where there is  a small dissipating force and a fraction of the total energy is removed after every oscillation and hence the amplitude decreases.

Critical damped – this is when there is an intermediate dissipating force and the system reaches equilibrium position as fast as possible without oscillating.

Over damped – there is a large dissipating force and the system takes longer to reach equilibrium position than critical damping. There are no oscillations in over damping.

4.3.3 State what is meant by natural frequency of vibration and forced oscillations.

Natural frequency – the frequency at which the system vibrates when in motion

Forced oscillation- when an external force is applied to the original frequency causing a change in the frequency of the oscillation.

4.3.4 Describe graphically the variation with forced frequency of the amplitude of vibration of an object close to its natural frequency of vibration.

The amplitude of the forced oscillation depends on  comparative values of the natural frequency and the driving frequency. In addition it also depends on the amount of damping present

Figure 4.3.1 – Variation with forced frequency of the amplitude of vibration of an object

4.3.5 State what is meant by resonance.

Resonance is when the natural frequency of a system is equal to the frequency of an external force. This results in oscillating and an increase in amplitude.

4.3.6 Describe examples of resonance where the effect is useful and where it should be avoided.

Useful

Quartz oscillators, microwave

Should be avoided

Generators and vibrations in machinery.

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