a level physics SHM

a pendulum oscillates backwards and forwards with a regular beat .

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

• 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

• 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

• 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)

• 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

• 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

• 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

• 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

Hz

the maximum displacement from the equilibrium position

change of displacement

gradient of the displacement time graph

• 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

the rate of change of velocity

the gradient of the velocity time graph

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

acceleration is negative

timing starts at max displacement

the centre of oscillation

• sin --> - cos --> -sin
• cos --> -sin --> -cos

radius

Acosθ

• velocity = distance / time 
• w = θ/t 
• wt = θ

Acos(wt)

2pift

Acos(2pift)

Acos(2pit/T)

• v = -2pifAsin(2pift) 
• v = -wAsin(wt)
• v = -2pi/TAsin(2pit/T)

radians mode

• a = -4pi²f²Acos(2pift) 
• a = -w²Acos(wt) 
• a = -4pi²/T²Acos(2pit/T)
• Acos(2pift) = x so
• a = -4pi²f²x  a = (2pif)²x
• a = -w²x
• a=-4pi²/T² x 

different periods and amplitudes

the oscillations could be out of step . to describe how far out of step oscillations are we use the idea of phase

how far through a cycle the movement is

2pi radias

the amplitude is small

• length of the pendulum 
• as the string gets longer the time period increases

mass of the bob

oscillates about the lowest/highest point

• mg=tcosθ vertically 
• -ma = tsinθ

• mg/cosθ = -ma/sinθ
• mgsinθ/cosθ = -ma 
• gtanθ= - a

• gtanθ = -a
• if θ small tanθ = sin θ 
• sinθ=x/l 
• gx/l = -a

restoring force wouldn't be proportional to displacement from equilibrium position because other forces now need consideration

• -(2pif)²x = -gx/l 
• -(2pif)²= g/l 
• 2pif = √g/l 
• f = √g/l / 2pi 
• f= 1/T
• T = 2pi√l/g

• measure T with a stopwatch 
• draw a graph of T² vs l 
• T² = 4pi²l/g 
• T² = 4pi²/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 = 4pi²/g

• 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

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

-kx

• a = f/m 
• a = -kx/m

• -(2pif)²x=-kx/m 
• -(2pif)²=k/m
• 2pif=√k/m 
• f = √k/m/2pi 
• T = 1/f 
• T = 2pi√m/k

a similar way to how we measured g for the simple pendulum the difference would be a graph of T² vs mass

k is increased or m os reduced

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

mg +kA to mg - KA where A = amplitude

compressed as much as possible when -A = x

the spring is stretched as much as possible when X = A

show the variation is sinusoidal

the masses must be identical

stationary for a moment so kinetic energy is zero

at the centre as you speed up towards this point

• at the extremes and lowest value at centre 
• so in moving from an extreme to centre potential energy is transferred to kinetic energy

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

max potential energy

• ep = 0.5fx
• f = kx
• ep = 0.5kx²

• 0.5kA²

0.5k(A²-x²)

• 0.5mv²=0.5k(A²-X²) 
• v²=k(A²-X²)/m 
• √k/m = 2pif 
• v = 2pif √A²-x² 
• when x = 0 vmax = 2pifA

parabolic in shape given by ep = 0.5kx²

• inverted parabola given by Ek = 0.5k(A²-x²) since kinetic energy is total energy - potential energy 
• 0.5kA² - 0.5kx²

always equal to 0.5kA² 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

kinetic energy max = 0.5mvmax² = 0.5m(2pifA)² = 0.5m4pi²f²A² =m2pi²f²A²

A²

free oscillations

the time period stays constant because both the amplitude and speed get smaller

air resistance

overcoming air resistance this effect is known as damping

oscillations take a while to die away

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

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

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

a pendulum moving through thick treacle

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

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

you push in time with the swings movement

an applied force at regular intervals

the frequency of the periodic force must match the natural frequency or resonant frequency of the oscillator

• applied frequency
• i.e. frequency of periodic force

amplitude builds up energy is transferred from the driver to the oscillator .

resonance

the larger the amplitude becomes

destructive

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

• pi/2 rad
• the driver leads the resonator by pi/2 rad

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

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

increase from pi/2 rad out of phase to pi rad out of phase (so from out of phase to anti phase)

the driver leads the resonator (oscillator)

below the amplitude of the driver

will increase and increase

will decrease more and more

damping

amplitude increases until the object is under too much pressure so breaks

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

resonance can be a problem

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

natural frequency if the damping is minimal

• 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

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

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

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.

the amplitude and frequency are the same

velocity

max kinetic and potential energy

two directions

1/4 of the time period

• speed is greatest
• displacement is least

• amplitude is greatest 
• velocity is least

a straight line

larger

• driving force is at same frequency as natural frequency of structure 
• resonance therefor occurs 
• large amplitude vibrations produced 
• could cause damage to structure

• stiffen the structure 
• increase damping by installing dampers