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why do old clocks use a pendulum
a pendulum oscillates backwards and forwards with a regular beat .

even as the oscillation dies away the time period
stays the same because . this is because the amplitude of the swing also decreases so the pendulum moves a smaller distance at a smaller speed hence the time period stays the same

SHM
 simple harmonic motion
 when an object oscillates with a constant time period even if the amplitude varies we say it's moving with SHM .
 also in SHM acceleration is proportional to the displacement of the object from equilibrium and is always directed towards the equilibrium position
 so in SHM acceleration acts in the opposite direction to displacement

velocity on a pendulum/swing
 at each end of the oscillation you are stationary for a moment
 the pendulum/swing speeds up as you move back to the centre
 once you pass this point you slow down again
 velocity is a vector quantity so we need to consider its direction we can take forward motion as positive and backward motion as negative

acceleration on a swing
 starting at 1 you are accelerating towards 3 (the equilibrium position)
 once you pass 3 you slow down until you stop at 5 . deceleration is the same as negative acceleration so decelerating away from 3 is the same as accelerating towards 3 . hence there's still acceleration towards the centre .
 from 5 you accelerate back towards the centre again
 from 3 to 1 you decelerate to a stop but this is the same as accelerating towards 1  the centre
 throughout the motion acceleration is directed towards the centre (equilibrium position)

how does the size of acceleration change
 when you pull the mass to one side and let it go it oscillates backwards and forwards
 when the mass is displaced to the left tension in k1 decreases and tension in k2 increases so the resultant pull of the strings is back to the right which which pulls the mass back to the centre (equilibrium position) each time
 newtons second law tells us the greater the force the greater the acceleration so the greater the force the greater the acceleration so the greater the displacement from the equilibrium position the greater the acceleration

how can we produce a displacement time graph for a simple pendulum
 using a pendulum pen
 as the pen swings back an forth it draws over the same line on the paper underneath
 if we pull the paper along steadily beneath the pen the pen would draw a regular wave
 this is the shape of the displacement time graph for SHM

The displacement time graph for SHM is
 sinusoidal . it is the shape of a sine or cosine curve
 displacement is a vector quantity so displacement in one direction is taken as positive and those in the opposite direction are taken as negative

one complete oscillation means
 a movement from one extreme to the other and back again
 the time this takes is called the time period , T
 the number of oscillations per second is called the frequency , f

frequency is measured in hertz
Hz

the amplitude of the motion is
the maximum displacement from the equilibrium position

velocity is the rate of
change of displacement

the velocity at any point in SHM is equal to the
gradient of the displacement time graph

if we started timing at the equilibrium position for a displacement time graph the gradient is maximum when
 displacement is zero , the gradient falls to zero as the swing reaches its maximum amplitude
 this tells us that the velocity is greatest when the displacement is zero and falls to zero at the extremes

acceleration is
the rate of change of velocity

the acceleration at any point in SHM is
the gradient of the velocity time graph

if we started timing at the equilibrium position for a velocity time graph the gradient is maximum when
the velocity is zero this is at maximum amplitude . so the acceleration is greatest at maximum amplitude , maximum amplitude is maximum displacement so acceleration increases as displacement increases

when the displacement is positive
acceleration is negative

if we start timing at the equilibrium position the displacement time graph is
velocity time graph is
acceleration time graph is

ignore side equations

we use cosine if the
timing starts at max displacement

we use sine if timing starts at
the centre of oscillation

displacement > velocity > acceleration
 sin >  cos > sin
 cos > sin > cos

using circular motion amplitude =
radius

using circular motion if we start at max displacement
x =
Acosθ

w =2pif
w = angular velocity
w =
 velocity = distance / time
 w = θ/t
 wt = θ




f = 1/t so x =
Acos(2pit/T)

velocity = differentiation of displacement so if we started max displacement v =
 v = 2pifAsin(2pift)
 v = wAsin(wt)
 v = 2pi/TAsin(2pit/T)

for these sorts of calculations we need to have our calc in
radians mode

acceleration = differentiation of velocity so if we start at max displacement a =
 a = 4pi^{2}f^{2}Acos(2pift)
 a = w^{2}Acos(wt)
 a = 4pi^{2}/T^{2}Acos(2pit/T)
 Acos(2pift) = x so
 a = 4pi^{2}f^{2}x a = (2pif)^{2}x
 a = w^{2}x
 a=4pi^{2}/T^{2} x

oscillations could have
different periods and amplitudes

however oscillations can have the same period and amplitude but differ
the oscillations could be out of step . to describe how far out of step oscillations are we use the idea of phase

phase measures
how far through a cycle the movement is

one complete oscillation has a phase of
2pi radias

a pendulum oscillates with SHM if
the amplitude is small

the time period of pendulum depends on
 length of the pendulum
 as the string gets longer the time period increases

the time period of a pendulum is independent of the
mass of the bob

if the bob is displaced from equilibrium then released it
oscillates about the lowest/highest point

at displacement x from the highest/lowest point when the thread is at an angle θ to the vertical the mass has two components
 mg=tcosθ vertically
 ma = tsinθ

restoring force = t
 mg/cosθ = ma/sinθ
 mgsinθ/cosθ = ma
 gtanθ=  a

provided θ is small
 gtanθ = a
 if θ small tanθ = sin θ
 sinθ=x/l
 gx/l = a

if θ greater than 10
restoring force wouldn't be proportional to displacement from equilibrium position because other forces now need consideration

a also = (2pif)^{2}x
 (2pif)^{2}x = gx/l
 (2pif)^{2}= g/l
 2pif = √g/l
 f = √g/l / 2pi
 f= 1/T
 T = 2pi√l/g

measuring acceleration due to gravity
T = 2pi√l/g
 measure T with a stopwatch
 draw a graph of T^{2 }vs l
 T^{2} = 4pi^{2}l/g
 T^{2} = 4pi^{2}/g * l
 y = m * x +c
 line of best fit should be a straight line through origin as no c so no y intercept
 m = 4pi^{2}/g

if we hang a mass on a spring and pull it down a small way and let it go the mass bounces up and down about its equilibrium it oscillates with sum what impacts the time period
 mass and stiffness
 the greater the mass the slower it accelerates under the same force this means the time period will increase
 a stiffer spring will pull the mass back to its equilibrium with more force , this produces more acceleration if the mass moves faster the time period will decrease

tension in spring = kΔl
t = kx > negative because the tension acts upwards trying to restore the object to its equilibrium



a also = (2pif)^{2}
 (2pif)^{2}x=kx/m
 (2pif)^{2}=k/m
 2pif=√k/m
 f = √k/m/2pi
 T = 1/f
 T = 2pi√m/k

we can calculate k in
a similar way to how we measured g for the simple pendulum the difference would be a graph of T^{2} vs mass

frequency of a mass on a spring is increased if
k is increased or m os reduced

time period in space doesn't depend on
g so to measure the mass of an astronaut they are placed between two springs and the time period is known since k is known we can calculate m

tension in the spring varies from
mg +kA to mg  KA where A = amplitude

minimum tension is when the spring is
compressed as much as possible when A = x

maximum tension occurs when
the spring is stretched as much as possible when X = A

how would you use a displacement time graph to show the motion is SHM
show the variation is sinusoidal

if T is the same for two different oscillations and the springs have the same constant
the masses must be identical

simple pendulum  at max amplitude you are
stationary for a moment so kinetic energy is zero

kinetic energy reaches a maximum
at the centre as you speed up towards this point

gravitational potential energy has a max value
 at the extremes and lowest value at centre
 so in moving from an extreme to centre potential energy is transferred to kinetic energy

mass oscillating on spring
energy changes from kinetic to elastic potential energy and back again every half cycle after passing through equilibrium

provided friction is constant , the total energy of the system is constant and is equal to
max potential energy

ep =
 ep = 0.5fx
 f = kx
 ep = 0.5kx^{2}


change in ep =
0.5k(A^{2}x^{2})

ep lost = ek gained
so
 0.5mv^{2}=0.5k(A^{2}X^{2})
 v^{2}=k(A^{2}X^{2})/m
 √k/m = 2pif
 v = 2pif √A^{2}x^{2}
 when x = 0 vmax = 2pifA

the potential energy curve is
parabolic in shape given by ep = 0.5kx^{2}

kinetic energy curve is an
 inverted parabola given by Ek = 0.5k(A^{2}x^{2}) since kinetic energy is total energy  potential energy
 0.5kA^{2}  0.5kx^{2}

the sum of the kinetic energy and the potential energy is
always equal to 0.5kA^{2} which is the same as the kinetic energy at zero displacement so the two curves add together to give a straight line for total energy

total energy =
kinetic energy max = 0.5mvmax^{2} = 0.5m(2pifA)^{2} = 0.5m4pi^{2}f^{2}A^{2 }=m2pi^{2}f^{2}A^{2}

total energy is therefor proportional to
A^{2}

so far we have assumed that no energy is lost from an oscillating system and that it continues to oscillate indefinitely , this is known as
free oscillations

but if you leave a pendulum or mass on a spring oscillating it eventually slows down and stops , but
the time period stays constant because both the amplitude and speed get smaller

what slows the oscillations down
air resistance

energy is lost in
overcoming air resistance this effect is known as damping


light damping is when
oscillations take a while to die away


example of heavier damping
imagine a pendulum moving through water , once released it would take longer to return to its equilibrium position and would hardly oscillate at all


critical damping
just enough to stop the system oscillating after it has been displaced and released from equilibrium


over damping
it doesn't oscillate after it has been displaced and takes a while to return to the equilibrium position

example of over damping
a pendulum moving through thick treacle

over damping is useful where
rapid fluctuations need to be ignored , an example of this is a car fuel gauge , over damping stops the pointer oscillating as the fuel sloshes in the tank

forced oscillations are
oscillations of a system that's subjected to an external periodic force

if you want to make a child's swing ho higher
you push in time with the swings movement

a periodic force is
an applied force at regular intervals

if you want amplitude to increase
the frequency of the periodic force must match the natural frequency or resonant frequency of the oscillator

driving frequency is
 applied frequency
 i.e. frequency of periodic force

if driving frequency matches applied frequency then
amplitude builds up energy is transferred from the driver to the oscillator .

this effect is called
resonance

the lighter the amplitude
the larger the amplitude becomes

resonance can be
destructive

cartoons sometimes show glasses smashing when somebody hits a high note . this can really happen
tje frequency of the sound (driving frequency) must match the natural frequency of the glass . the glass then resonates , vibrating more and more until it breaks

the phase difference between the periodic force and displacement is always ....when resonance occurs in other words ...
 pi/2 rad
 the driver leads the resonator by pi/2 rad

this means that
the periodic force is exactly in phase with the velocity of the oscillating object

as the applied frequency (driving frequency) increases towards the natural frequency
the phase relationship between displacement and periodic force increases from 0 to pi/2 rad so from in phase to pi/2 rad out of phase

as the applied frequency (driving frequency) increases past the natural frequency the phase relationship between displacement and periodic force
increase from pi/2 rad out of phase to pi rad out of phase (so from out of phase to anti phase)

when there's a phase difference
the driver leads the resonator (oscillator)

amplitude frequency graph

below the natural frequency the amplitude of the oscillation will be
below the amplitude of the driver

at the natural frequency the amplitude of the oscillation
will increase and increase

above the natural frequency the amplitude of the oscillations
will decrease more and more

damping reduces the effect of
damping

without damping the
amplitude increases until the object is under too much pressure so breaks

as damping increases
 amplitude of the resonance increases
 resonance peak gets broader
 resonant frequency is slightly lower than natural frequency

damping is used where
resonance can be a problem

a good example is the damping of
buildings in earthquake zones . the foundations are designed to absorb energy . this stops the amplitude of the buildings oscillations reaching dangerous levels when an earthquake arrives

resonant frequency is only equal to
natural frequency if the damping is minimal

explain the formation of unpleasant noises coming from a loudspeaker
 a loudspeaker vibrates in response to the oscillating electrical signal that drives it , thus it undergoes forced vibrations
 forced vibrations occur at the driving frequency of electrical signals
 if the driving frequency matches the natural frequency of the loudspeaker then large amplitude vibrations occur

tacoma narrows bridge
frequency of the forces from the wind matched natural frequency of the bridge and large amplitude oscillations built up destroying the bridge a similar thing happened in frame however the frequency of the marching soldiers matched the natural frequency of the bridge

how do microwaves work
the frequency of microwaves almost equals natural frequency of vibration of water molecules . this means the water molecules in food resonate . this means they take in energy from the microwaves and get hotter , this heats the food , there is a slight mismatch in frequencies it prevents all the energy being absorbed at the surface and allows microwaves to penetrate deeper in the food

what happens if x is displaced and released so it oscillates in the plane perpendicular to the plane of the pendulum at rest
the effect of the oscillating motion of x is transmitted along the support thread , subjecting each of the other pendulums to forced oscillations . Pendulum D will oscillates with the largest amplitude .Pendulum X and D have equal length and consequently equal natural frequency. Therefore resonance happens to pendulum D, and it oscillates with maximum amplitude.

at any point in a progressive wave
the amplitude and frequency are the same

damping force is always in the opposite direction to ...
velocity

damping reduces
max kinetic and potential energy

displacement and amplitude have
two directions

to travel from displacement to the centre of oscillation is
1/4 of the time period

acceleration is least when
 speed is greatest
 displacement is least

acceleration is greatest when
 amplitude is greatest
 velocity is least

in sum the graph of acceleration vs displacement is
a straight line

time period of a pendulum on the moon would be .... than the time period of the same pendulum on earth
larger

To celebrate the Millennium in the year 2000, a footbridge was constructed across the River
Thames in London. After the bridge was opened to the public it was discovered that the structure
could easily be set into oscillation when large numbers of pedestrians were walking across it.
(b) Under what condition would this phenomenon become particularly hazardous? Explain your answer.
 driving force is at same frequency as natural frequency of structure
 resonance therefor occurs
 large amplitude vibrations produced
 could cause damage to structure

Suggest two measures which engineers might adopt in order to reduce the size of the
oscillations of a bridge
 stiffen the structure
 increase damping by installing dampers

