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Simple Harmonic Motion.


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Revision.

For this part of the course you need to be

familiar with the concepts of circular motion and

angular velocity.

State the formulas used to calculate angular

velocity.


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SHM

A system will oscillate if

there is a force acting on it

that tends to pull it back to

its equilibrium position a

restoring force.

!n a swinging pendulum the

combination of gravity and

the tension in the string that

always act to bring the

pendulum back to the centre

of its swing.

"ension

#ravity

Resultant$restoring

force%


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For a spring and mass the

combination of the gravity

acting on the mass and

the tension in the spring

means that the system willalways try to return to its

equilibrium position. M

"

g

Restoring

force


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!f the restoring force f is directly proportional to the

displacement x & the oscillation is known as simple

harmonic motion $SHM%.

For an ob'ect oscillating with SHM

f ∝ -x

The minus sign shows that the restoring force is acting

opposite to the displacement.

Acceleration !n SHM


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(isplacementRestoring

force


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)inetic and *otential energy in SHM


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+e already know that f=ma, we can substitute this into

the above formula to give.

a ∝ -x

!f we put in a constant we get the equation

a = -ω 2 x

ω, is a constant& ω is called the angular velocity and isdependant on the frequency of the oscillations and can

be written asω

=2 π

f.

So the equation for SHM can be written as-

a = -(2 π f)2

x


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uestions.

/. 0se the equation a = -ω 2x to work out the correct unitsof ω .

,. Sketch an oscillating pendulum and mark in the

positions of where the acceleration is greatest andsmallest.

1. An ultrasonic welder uses a tip that vibrates at ,2 kH3

if the tip4s amplitude is 5.,26/78, calculate thema6imum acceleration of the tip


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9. A wave has a frequency of 9 H3 and an amplitude of7.1m calculate its ma6imum acceleration.

2. A system is oscillating at 177 kH3 with an amplitude of7.5 mm calculate its ma6imum acceleration.

5. A speaker cone playing a constant bass note is

oscillating at /77 H3& the total movement of thespeaker cone is / cm. Assuming that the movement isSHM calculate its ma6imum acceleration.


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(isplacement.

!f a system oscillates with SHM the pattern of the

motion will be the same& the motion will follow

the same rules.

"he equation for the displacement& x & isrelated to the time& t & by the equation-

x = Acosω t (remember to use radians)


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+e already know that ω can also be written as

2 π f, then the equation can also be written as-

x = Acos2 π ft

+e can use this equation to predict the positionof an oscillating system at any time.


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s

A pendulum is oscillating at 17 times per minute

and has an amplitude of ,7 cm& find itsamplitude 7.2s after being released from itsma6imum displacement.

Find its displacement 7.:2s after being releasedfrom its ma6imum displacement.

!f the pendulum is oscillating at /,7 times perminute and has an amplitude of 17 cm.

+ork out the displacement at 7., s& 7.9 s and7.2 s


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!f high tide is at /, noon and the ne6t is /, hours

later and the amplitude of the tide is ,m we can

work out the height of the tide at any time for

e6ample ,pm.

Find out the height of the tide ,pm later $9pm%.


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;nergy in simple harmonic motion.

+e already know that the total energy$mechanical energy% for a system that is movingwith SHM is the sum of the potential and kineticenergies.

+hen the ob'ect is at the e6tremes of itsoscillation $6 < ± A% it has no ); but the *; is at

its ma6imum. +hen the ob'ect is mid waythrough its oscillation $when 6 < 7% the ); is atits ma6imum but there is no *;.


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"he potential energy is equal to the work done.

;p < = mω,

6,

At the ma6imum displacement ; < ;p so

; < = mω, A,

And

= mω, A, < = mω,6, > = mv,

!f we divide = m.

ω, A, < ω,6, > v,


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So

v, < ω,$A,8 6,%

or

v < ±,πf √ A,86,

+e can use this equation to workout the velocity

of an ob'ect at any position in it4s SHM.


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A / kg mass is hanging from a spring which has a spring

constant $k% of /777 ?m8/& it is oscillating with an amplitude

of ,cm.

a% 0se " < ,π √ m@k to calculate the time period of theoscillation.

b% use v< ±,πf √ A,86, to find the velocity and then thekinetic energy of the mass.

Displacement,

x (m)

Velocity, (m/s) KE, (joules)

7.7,

7.7/27.7/

etc

87.7,


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s

A metal strip is clamped to the edge of the tableand has an ob'ect of mass ,7g attached to thefree end. "he ob'ect is pulled down andreleased. "he ob'ect vibrates with SHM with anamplitude of .7 cm and a period of 7./5 s.

Balculate the ma6imum acceleration of theob'ect

Balculate the ma6imum force

State the position of the ob'ect when it has no);.


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(escribing SHM


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http-@@physics.bu.edu@Cduffy@semester/@

c/DSHMDgraphs.html

http://physics.bu.edu/~duffy/semester1/c18_SHM_graphs.htmlhttp://physics.bu.edu/~duffy/semester1/c18_SHM_graphs.htmlhttp://physics.bu.edu/~duffy/semester1/c18_SHM_graphs.htmlhttp://physics.bu.edu/~duffy/semester1/c18_SHM_graphs.html


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(isplacement

$6%

Acceleration $a%

π out of phase$/7 deg%

Eelocity $v%

π@, out of phase$7 deg%


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Simple pendulum

A pendulum consists of asmall Gbob of mass m&suspended by a lightine6tensible thread of

length l & from a fi6ed point.

"he bob can be made tooscillate about point I in a

vertical plane along the arcof a circle.

+e can ignore the mass ofthe thread


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+e can show that oscillating simple pendulums

e6hibit SHM.

+e need to show that a x.Bonsider the forces acting on the pendulum-

weight& W of the bob and the tension& T in the

thread.

+e can resolve + into , components parallel

and perpendicular to the thread-

*arallel- the forces are in equilibrium

*erpendicular- only one force acts& providing

acceleration back towards I


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*arallel to string-

F < mg cosθ

*erpendicular- to string

F < restoring force towards I

< mg sinθ +e already know that F < ma

So F< 8mg sinθ < ma$8ve since towards I%


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At small angles $θ < less than /7 deg%

Sinθ is appro6imately equal to θ $in radians%

θ appro6imates to 6@l for small angles.

So: ∴ 8mg $ x @l% < ma

Rearranging- a < 8g $ x @l% < pendulum equation

$can also write this equation as a < 8 x $g@l%%


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!n SHM a ∝ x SHM equation a < 8$,πf %, x

*endulum equation a < 8 x $g@l%

Hence $,πf %, < $g@l%

∴ f < /@,π $√g@l% $remember " < /@f)

" < ,π $√l@g%

"he time period of a simple pendulum dependson length of thread and acceleration due to

gravity


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;6periment

0sing a long clamp stand& a pendulum bob&

some light string and a stop watch to

investigate the relation ship between g& l

and "

For a pendulum of known length count the

time taken for /7 complete oscillations

$there and back%.

0se the pendulum formula to calculate the

force of gravity.

Repeat with 1 other lengths.


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SHM in Springs


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!n a spring8mass system.

(o you think the si3e of the mass affects the

"ime *eriod of the IscillationJ +hat do you

think the relationship will beJ

(o you think the stiffness $spring constant% of

the spring will affect the "ime *eriod of the

IscillationJ +hat do you think the relationship

will beJ


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"he diagram here shows a mass8

spring system.

Set the equipment up and use

F


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How did the different masses affect the period of

oscillation.

How did the spring constant $different

arrangements of springs% affect the period of

oscillationJ


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"he time period of oscillation of a spring is

dependant on the spring constant of the spring

and the mass of the system.!t is independent of the force of gravity.

"he relationship is.

= k m

T π 2


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/ Balculate the time period of a spring masssystem of ,.2 kg with a spring const. of ,77?@m.

, +hat is the frequency of a ,7g mass oscillatingon the end of a spring with a const. of /,7 ?@m.

1 A spring is oscillating 92 times per min.calculate the mass if the const. is /777 ?@m.

9 !f a mass of /7777 )g is oscillating at afrequency of 7.1: H3 what is the const. of thespringJ


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Resonance and damping.


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Resonance.

Resonance is the tendency in a system tovibrate at its ma6imum amplitude at a certainfrequency. "his frequency is known as the

systemKs resonance frequency. +hen dampingis small& the resonance frequency isappro6imately equal to the natural frequency ofthe system& which is the frequency of freevibrations.

"he natural or fundamental frequency is oftenwritten as f


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*erhaps one of the more common e6amples of

resonance is in musical instruments. For

e6ample in guitars it is possible to make other

strings vibrate Gsympathetically when another is

plucked& either at their fundamental or overtone

frequencies.

http-@@www.youtube.com@watchJv


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;6amples of resonance.

*ushing a child on a swing ma6imum A whenpushing ƒ < ƒo

"uning a radio electrical resonance occurswhen ƒo of tuning circuit ad'usted to match ƒ ofincoming signal

*ipe instruments 8 column of air forced tovibrate. !f reed ƒ < ƒo of column loud sound

producedRotating machinery e.g. washing machine. Anout of balance drum will result in violentvibrations at certain speeds


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"acoma narrows bridge.

"he "acoma narrows bridge is often used as ane6ample of resonance& although it is not strictlyscientifically accurate to do so. !t does how evergive an e6ample of what can happen if an ob'ect

was to be kept at its resonant frequency for along time.

http-@@www.youtube.com@watchJv


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Larton4s *endulum

All ob'ects have a natural frequency of vibrationor resonant frequency. !f you force a system 8 inthis case a set of pendulums 8 to oscillate& youget a ma6imum transfer of energy& i.e. ma6imum

amplitude imparted.

+hen the driving frequency equals the resonantfrequency of the driven system. "he phase

relationship between the driver and drivenoscillator is also related by their relativefrequencies of oscillation.


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Qou also get a very clear illustration of the phaseof oscillation relative to the driver. "he pendulum

at resonance is @, behind the driver& all theshorter pendulums are in phase with the driverand all the longer ones are out of phase.

"he amplitude of the forced oscillations dependon the forcing frequency of the driver and reacha ma6imum when forcing frequency < naturalfrequency of the driven cones.


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Another e6ample of resonance in a driven

system is the hacksaw blade oscillator.

(riving mass

And arm

Slave4 arm withslave4 mass

;lastic

band

pointer


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!f we change the period of oscillation of the

driver by moving the mass $increasing T% the

hacksaw blade will vibrate at different rates& if

we get the driving frequency right the slave will

reach resonant frequency and vibrate wildly.

!f we move the masses on the blade it will have

a similar effect.


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(amping

"he amplitude depends on the degree of damping


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A damped spring

Set up a suspended mass8spring system with a

damper4 a piece of cardattached hori3ontally to themass to increase the airdrag. Alternatively& clamp a

springy metal blade $e.g.hacksaw blade% firmly tothe bench. Attach a massto the free end& and add a

damping card.

Show how the amplitudedecreases with time.


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