Section IV 14 Oscillations

7/28/2019 Section IV 14 Oscillations

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Oscillations

14. Oscillations Content

14.1 Simple harmonic motion

14.2 Energy in simple harmonic motion

14.3 Damped and forced oscillations: resonance

Learning Outcomes (a) describe simple examples of free oscillations.

* (b) investigate the motion of an oscillator using experimentaland graphical methods.

(c) understand and use the terms amplitude, period, frequency,

angular frequency and phase difference and express theperiod in terms of both frequency and angular frequency.

(d) recognise and use the equation a =2x as the definingequation of simple harmonic motion.

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(e) recall and use x = xo sin t as a solution to the equation a = 2

x. (f) recognise and use v = vocos t, v = (x2

o x2)

* (g) describe with graphical illustrations, the changes in displacement,velocity and acceleration during simple harmonic motion.

(h) describe the interchange between kinetic and potential energyduring simple harmonic motion.

* (i) describe practical examples of damped oscillations with particular

reference to the effects of the degree of damping and the importance ofcritical damping in cases such as a car suspension system. (j) describe practical examples of forced oscillations and resonance. * (k) describe graphically how the amplitude of a forced oscillation

changes with frequency near to the natural frequency of the system,and understand qualitatively the factors which determine the frequencyresponse and sharpness of the resonance.

(l) show an appreciation that there are some circumstances in which

resonance is useful and other circumstances in which resonanceshould be avoided.

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Oscillations and vibrations

Vibrations and oscillations occur all the time and are everywhere.

Vibrations are physical evidence of waves, such as a loud stereo shaking a

table, i.e. sound waves cause vibrations

One complete movement from the starting point or rest point or equilibrium

position and back to the starting point or rest position or equilibrium positionis known as an oscillation

The time taken for one complete oscillation is referred to as the periodT ofthe oscillation

The number of oscillations per unit time is the frequency f

Frequency f = 1/T , may be measured in hertz (1 Hertz = 1 s-1) or in min-1,hour-1 etc

The distance from the equilibrium position is known as the displacement andit is a vector quantity since the displacement may be on either side of the

equilibrium position

The amplitude(a scalar quantity) is the maximum displacement 3


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Examples of oscillatory motion

beating of a heart

a simple pendulum

a vibrating guitar string

vibrating tuning fork

atoms in solids

air molecules oscillate when sound waves travel through air.

oscillations in electromagnetic waves such as light and radio waves

oscillations in alternating current and voltage.

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Recap from study of waves

Some oscillations maintain a constant period even when the amplitudeof the oscillation changes. This is known as isochronous and has beenmade use of in timing devices

Galelli Galileo discovered this for a pendulum. A pendulum swingingwith a large amplitude is not isochronous

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Displacement-time graphs

It is possible to plot displacement-time graphs for oscillators

The graph describing the variation of displacement with time may have

different shapes depending on the oscillating system

For many oscillators the displacement-time graph of a free oscillationis approximately a sine or cosine curve

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Simple harmonic motion (shm)

A sinusoidal displacement time graph is a characteristic of animportant type of oscillation called simple harmonic motion(shm)

In harmonic oscillators the amplitude is constant with time

SHM is defined as the motion of a particle about a fixed point suchthat its force F or acceleration a is proportional to its displacement x

from the fixed point, and is directed towards the point F is known as the restoring force

Mathematically it is defined as a = - 2x where is the angularfrequency and is equal to 2f

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cont The defining equation is represented in a graph of a against x as a

straight line ofnegative gradient through the origin.

Gradient is negative because of the minus sign in the equation whichrepresents that acceleration is always directed towards the fixed pointfrom which the displacement is measured

This means that in shm, acceleration is directly proportional to thedisplacement/distance from the fixed point and is always directed tothat point

Acceleration is always opposite to the displacement since the force isalso opposite to the displacement

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a

x0


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Comparisons

In linear motion, acceleration is constant in magnitude and direction

In circular motion acceleration is constant in magnitude but notdirection

In simple harmonic motion the acceleration changes periodically inmagnitude and direction

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Solution of equation for shm

In order to find the displacement time relation for a particle moving in

shm, we need to solve the equation a = - 2x which requiresmathematics beyond the requirements of A/AS

However we need to know the form of the solution

x = x0 sint or x = x0 costwhere x0 is the amplitude of the oscillation

The solution x = x0 sint is used when at time t= 0, the particle is atits equilibrium position wherex = 0, and conversely if at time t= 0 the

particle is at its maximum displacement,x = x0 the solution is x = x0

cost

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Velocity & acceleration for shm

The velocityvof the particle is given by the expressions

v = x0 cost when x = x0 sint

v = -x0 sint when x = x0 cost

The maximum speed is given by v0= x0

An alternate expression for the velocity is v =(x02x2)

(which will be derived next)

The accelerationaof the particle is given by the expressions

a = -x02sint when x = x0 sint

a = -x02cost when x = x0 cost

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Displacement, velocity and acceleration graphs

x

v

a

t

t

t

Displacement (x), velocity (v) & acceleration time graph


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Alternate expression for velocity

Recall that x = x0 sint and v = x0 cost

So sint = x/x0 and cost = v/(x0)

Trigonometric relationship between sine and cosine is

sin2 + cos2 = 1

Applying the above relationship, we have

x2/x02 + v2/(x0

22) = 1 which gives

v2 = x022 - x2 2 , hence

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v = (x02 - x2)


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Example

The displacement xat time t of a particle moving in shm is given by x =0.25 cos 7.5t where x is in metres and t is in seconds.

a) use the equation to find the amplitude, frequency and period for the

motion

b) find the displacement when t = 0.50 s

Solution

a) Compare the equation with x = x0 cost

The amplitude x0 = 0.25 m, = 2f = 7.5 rad/s, therefore f = 1.2 Hzand period T = 1/f = 0.84 s

b) Substitute t = 0.50 s in the equationt = 7.5 x 0.50 = 3.75 rad = 215

so x = 0.25 cos 215 = -0.20 m

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Worked examples of shm

Mass on a helical spring

Simple pendulum

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Mass on a helical springHookes law

Consider a mass msuspended from a spring The weight mg is balanced by the tensionT in the spring

When the spring is extended downwards by an amount x away fromthe equilibrium position, there is an additional upward force called the

restoring forcein the spring given by F = - kx

When the mass is released the restoring force F pulls the massupwards towards the equilibrium position. The minus sign shows the

direction of this force.

As the force is proportional to the displacement, the acceleration is

also proportional to the displacement and is directed towards the

equilibrium position meeting the condition for shm

The full theory shows that the period of oscillationT = 2(m/k)

since F = ma, then ma = - kx

hence a = - (k/m)x = dv/dt = d2x/dt2

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The simple pendulum

A simple pendulum is a point mass m on a light inelastic string although in realexperiments we use a finite pendulum bob of finite mass

When the bob is pulled aside through an angle and released, there will be a restoringforce acting in the direction of the equilibrium position

Because the pendulum moves in an arc of a circle, the displacement will be an angular

displacement rather than a linear displacement

The 2 forces on the bob are its weight mgand the tensionT in the string

The component of the weight along the direction of the string mg cos, is equal to thetensionT in the string

The component of the weight at right angles to the direction of the string, mg sin , isthe restoring force F.This makes the bob accelerate towards the equilibrium position

The restoring force depends on . As increases the restoring force is notproportional to the displacement and so the motion is oscillatory but not shm, but if the

angle is kept small (less than 5), is proportional to sin and exhibits shm (checkusing your calc)

The full theory shows period of oscillationT = 2(l/g) where g is the acceleration offree fall

A simple pendulum can be used to experimentally determine g by repeating the

experiment with different lengths of pendulum and plotting a graph ofT2

against 42

/g 17


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Hookes law & shm

Any system which obeys Hooke's Law exhibits shm

but i) extensions must not exceed the limit of proportionality

ii) the spring must have small oscillations as large amplitudeoscillations may cause the spring to become slack

iii) the spring should have no mass; if the mass is > 20x themass of the spring, the error is 1%

This example of shm is a particularly useful model for interatomic forces

and vibration of molecules containing atoms oscillating as if connected by

tiny springs

The frequency of oscillation can be measured using spectroscopy whichgives direct information about the bonding

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Example

A light spring of spring constant k hangs vertically from a fixed point and a mass misattached to its free end.

a) State 2 conditions that must be met before the subsequent motion may beconsidered to be simple harmonic.

b) Derive an expression for the period T of the resulting motion.

Solution

a) 2 conditions for shm are:

a) The equilibrium position due to the mass is within the Hookes law limit of the spring b) the mass is given a small vertical displacement such that the springs Hookes law

limit is not exceeded

b) Let x = displacement of mass m, a = acceleration of mass m, F = ma = -kx

Force in a spring is, - kx = ma , hence a = - (k/m)x

As a is proportional to - x , so resulting motion is shmi.e. a = - 2x

a = - (k/m)x = - 2x, so angular frequency = (k/m)Therefore period T =2/ =2(m/k)

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ExampleA light string of length l hangs vertically from a fixed support and a mass m is attached to itsfree end. The mass is given a horizontal displacement and released to swing freely.

a) State a condition which must be satisfied before the resulting oscillation may be

considered shm.

b) Derive an expression for the period T of the resulting motion.

Solution

a) A required condition is that the angular displacement is small l

b) Let x = displacement of mass m, a = acceleration of mass m

In the direction perpendicular to string, F = ma

- mg sin = ma, so - g sin = a

For small , sin x/l, so - gx/l a x

As a is proportional to - x , so resulting motion is shm

i.e. a = - 2x

Hence, - (g/l)x = - 2x, so = (g/l) mg

Therefore period T = 2/ = 2(l/g)

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Example

A helical spring is clamped at one end and hangs vertically. It extends by 10

cm when a mass of 50 g is hung from its free end.

Calculate: a) the spring constant of the spring

b) the period of small amplitude oscillations of the mass

Solutiona) k = F/x, k = 4.9 Nm-1

b) T = 2(m/k) T = 0.63 s

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Energy changes in shm

A system exhibiting simple harmonic motion would possess a constant totalenergy at all points of time

The total energy normally comprises a portion of potential energy andanother balanced portion of kinetic energy.

There is thus a continuous interchange of the two energies duringoscillations.

For example, a weighted helical spring has a total energy that is the sum ofthe kinetic energy of the moving mass and the stored elastic potentialenergy of the spring.

Plotting on the same graph for energy versus time/displacement, the twosinusoidal curves are completely out of phase.

It can be proven that the total energy of a weighted spring is m2

xo2

which is a constant.

.

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Energy vs time graph

energy

Energy versus time graph

0

0 T/4 T/2 3T/4 T time

total energyK.E. P.E.


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Displacement, velocity and acceleration graphs

x

v

a

t

t

t

Displacement (x), velocity (v) & acceleration time graph


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Energy changes in shm

The kinetic energy of a particle of mass m oscillating with shm ismv2and from the earlier derivation v2= x022- x22

So k.e Ek at displacement x is m2(x02- x2) To find the potential energy Epwe need to find the work done against

the restoring force;

since F = ma , Fres = - m2x but averagerestoring force = m2x Hence work done = average restoring force x displacement

= m2x2

The total energy Etotof the oscillating system is given by

Etot= Ek+Ep = m2(x02- x2) + m2x2

= m2x02

This total energy is constant as it merely expresses the law ofconservation of energy

Pg 272 Chris Mee figs 10.22, 10.23,10.24

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Example

A particle of mass 60 g oscillates with shm with angular frequency of 6.3

rad/s and amplitude 15 mm.

Calculate a) the total energy

b) the k.e and p.e at half amplitude (i.e. at x = 7.5 mm)

Solution

Etot= Ek+Ep = m2(x02- x2) + m2x2= m2x02

a) Etot= m2x0

2 = 2.7 x 10-4 J

b) Ek= m2(x0

2- x2) = 2.0 x 10-4 J

Ep = m2x2 = 0.7 x 10-4 J

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Natural frequency & resonance

A particle is said to beundergoing free oscillations when the only

external force acting on it is the restoring force

The total energy remains constant at all points of time

A free oscillation is one where an object or system oscillates in theabsence of any damping forces, and it is said to oscillating in itsnatural frequency

In real situations, frictional and other resistive forces cause theoscillators energy to be dissipated, and this energy is convertedeventually into heat energy. The oscillations are said to be damped

When one object vibrates at the same frequency as another it is saidto be in resonance

The swing of a frictionless pendulum is an example of a freeoscillation.

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Resonance

In the absence of external forces to an oscillating system, the system oscillates at its

natural frequency f0. The only forces acting are the internal forces of the oscillating

system

When an external force is applied to an oscillating system, the system is under forcedoscillations and will vibrate at the frequency of the applied force rather than at thenatural frequency of the system

Whether or not the forcing frequency equals the natural frequency, the oscillations aresaid to be forced when a periodic force acts.

When the forcing frequency is equal to the natural frequency, net energy is taken in

and the amplitude of oscillation builds up further and the applied periodic force is said to

have set the system in resonance. Under such condition, further resonance will result in

more energy being taken in to build up the amplitude further.

Resonance occurs when a system is forced to oscillate at its natural frequencyby thedriving frequency

When resonance occurs, the amplitude of the resulting oscillations is a maximum asmaximum energy is transferred from the forcing system

E.g. Barton's pendulumonly the pendulum with the same length as the original willoscillate with the biggest amplitude

Applicationswind instruments, excessive noise from a moving bus, radio & tv tuning

The Tacoma Narrows suspension bridge in Washington State, USA in 1940 collapsed due

to a moderate gale (of same frequency as natural frequency of bridge) setting the bridge

into resonance until the main span broke up 28


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Damped oscillations

A damped oscillationis one where frictional forces present

gradually slow down the oscillation and the amplitude decreaseswith time i.e. decreasing energy

Damped oscillations are divided into under-damped, criticallydamped and over-damped oscillations

An under-damped(lightly damped) oscillation is one where theamplitude of oscillation or displacement of the system decreaseswith time. Example: oscillation of a simple pendulum with thedamping or dissipative force as air resistance

In a critically dampedsystem, oscillations are reduced to zero in theshortest possible time. Examples: moving coil ammeter or volt

meter, shock absorber, door closer

In an over-damped(heavy damping) system, a displacement from itsequilibrium position takes a long timefor the displacement to bereduced to zero. Example: door dampers

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Damped oscillations

Displacement vs Time Graph

x

t

under-damped critically-damped

over-damped


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Effects of damping on forced oscillations

Pg 277 Chris Mee fig 10.29 & 10.30

As the degree of damping increases:

The amplitude of oscillation at all frequencies is reduced

The frequency at max amplitude shifts gradually towards lowerfrequencies

The peak becomes flatter

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Electrical resonance

Electrical oscillators made from combinations of capacitors and

inductors(coils) can also be forced into oscillations or be made to resonate

This is the basis of tuning in electronic circuits which pick out therequired transmission in a receiver

The natural frequency of an electrical oscillator depends on the capacitorand inductance of the coil used. By varying the capacitance, we can tune in

to different channels

The range of frequencies selected depends on the damping which in turn

depends on the resistance in the circuit

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Section IV 14 Oscillations

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