Chapter 17 - Oscillations and SHM Flashcards
define oscillatory motion
“oscillating motion occurs where an object starts in an equilibrium position and a force is applied, causing it to oscillate either side of its equilibrium position”
explain the ‘phases’ of oscillatory motion
- an object is displaced from its equilibrium position
- it accelerates towards the equilibrium position and decelerates once it’s passed it
- it reaches its maximum displacement the other side of the equilibrium position and the process repeats
define displacement and amplitude, give their symbols and units
Displacement - x/m - Distance with respect to direction from its equilibrium position
Amplitude - A/m - Maximum displacement from equilibrium position
define period, frequency and phase difference, give symbols and units
Period - T/s - time taken to complete one full oscillation
Frequency - f/Hz - number of complete oscillations per unit time
Phase difference - Phi/ radians - the difference in phase of the cycles of the oscillations
define angular frequency
“angular frequency describes the motion of an oscillating object through the rate of change of angle”
angular frequency = 2(pi)/period
what are the equations for angular frequency
Omega = 2(pi)/t
Omega = 2(pi)f
give the equation for SHM and identify the main points
a = (-)(omega^2) x
where omega^2 is constant:
- acceleration is directly proportional to displacement
- it acts in the opposite direction (towards the equilibrium position)
- maximum acceleration occurs at maximum displacement (amplitude)
give the three main features of SHM
- an acceleration-displacement graph has a constant gradient pf Omega^2
- frequency and period are constant
- period of a pendulum in SHM is independent of amplitude
describe the practical used to determine the frequency and period of an object in SHM
- set up a pendulum on a clamp stand or a spring with masses attached to a clamp
- pick a point as a fiducial marker
- set the pendulum swinging or spring oscillating and time n oscillations using the fiducial marker to help
- divide the time by n to calculate the period of an oscillation
- repeat for different amplitudes
- calculate an average
what shape are the graphs for SHM (displacement against time)
- sinusoidal
- if there are no energy losses Amplitude is constant
what are the two possible shapes of a displacement-time graph for SHM
- if starting from max displacement then cosine shape
- if starting from equil position then sine shape
assuming we start from max displacement what is the shape of the velocity time graph for SHM
- the derivative of cosine = -sine
- so a -ve sine graph
assuming we start from max displacement, what is the shape of the acceleration time graph for SHM, what is important about this
- the double derivative of Cosine = -cosine
- so a is directly proportional to -x
what are the equations linking displacement and time for SHM
X = Acos(omega*t)
- if starting from max displacement
- if ans is -ve then it is on the other side of the equil position
X = Asin(omega*t)
- if from equil position
- assign one side +ve and one side -ve
what is the equation for SHM linking velocity and displacement, amplitude and omega, what is the Vmax equation
V = +- omega Sqrt(A^2 - X^2)
Vmax = omegaA
- Because max velocity occurs at equil position (X=0)
define damping
“an oscillation is damped when an external force that acts on the oscillator has the effect of reducing the amplitude of its oscillations”
what are the key points about light damping, give an example
light damping:
- low energy losses
- period almost unchanged
- amplitude gradually decreases
e.g. air
what are the key points of heavy damping and give an example
heavy damping:
- higher energy losses
- period increases slightly
- amplitude decreases significantly
e.g. water
what are the key points of very heavy damping and give an example
Very heavy damping:
- no oscillatory motion
- slowly returns to equil position
e.g. syrup
what generally happens to the kinetic energy of the oscillator in damping
KE —> other forms (usually heat)
what do the graphs of the types of damping look like
- light damping = sinusoidal but decreasing amplitude (exponentially)
- heavy damping dips below the x axis comes slightly above then back to rest
- very heavy damping looks like a -ve exponential
what is a free oscillation
“a free oscillation is where a mechanical system is displaced from its equilibrium position and oscillates without external forces”
frequency = natural frequency
what is a forced oscillation
” a forced oscillation is one in which a periodic driver force is applied to an oscillator”
freq = freq of driver
what is resonance
“Resonance occurs where the driving frequency is equal to the natural frequency of the object, this causes amplitude to increase dramatically”
“Amplitude at this point is at a MAXIMUM”
describe Barton’s pendulum experiment
- a wire is fixed across two points
- D is a heavier brass bob pendulum
- there are a number of different lengths paper cone pendulums
- D is set swinging, acting as a driving force
- if a pendulum has the same length as D it will resonate and have a much greater amplitude than the other pendulums
what can we say about the energy in a system of an object moving with SHM
- total energy remains constant assuming no energy losses
describe the energy transfers that occur in an oscillation of a pendulum
- at the amplitude the object has no KE, only potential energy, in this case GPE
- as the pendulum falls it loses GPE but gains KE
- as it moves through the equilibrium position, it has maximum KE and no GPE
what are the differences in the sources of potential energy between a pendulum and a mass-spring system
if the mass-spring system is vertically:
- potential energy is GPE and Elastic Potential
if the mass-spring system is horizontally
- potential energy is only Elastic Potential
what does the graph of energy against displacement look like for an object in SHM
- two parabolas
- one -ve, one +ve
- the +ve shaped parabola is Ep such that at max displacement, Ep is max
- the -ve shaped parabola is KE such that at max displacement, KE = 0
- total energy remains constant as a line running across the top
explain a simple demonstration of energy transfer in SHM
- spring and glider on an air track
- compress spring
- Elastic potential = 1/2 k X^2
- this is transferred to the kinetic energy of the glider (1/2 mv^2)
how can the air track demonstration explain the shape of the graph
at any point
KE = total energy - Ep
total energy = Ep max
Ep max = 1/2 k A^2
so
KE = 1/2 k A^2 - 1/2 k X^2
KE = 1/2 k (A^2-X^2)
so both KE and EP have parabola shapes
what do the Ep and KE against time graphs look like against time for a pendulum
if pendulum starts from 0 displacement:
- Ep-time graph is like sin but only positive so Mod(sin)
- KE-time graph is like V^2 so cos^2, in other words a cos graph shifted up
give some examples of where resonance is used
- clocks use the resonance of a pendulum or quartz crystal to keep time
- musical instruments have resonant casings and the air column inside instruments resonates
- tuning circuits use resonance e.g. car radios
- MRI scanners
what does the graph of Amplitude of oscillation against driving frequency look like, give its key features
- a number of lines which rise up to a peak then drop away but not a smooth parabola
- the lines represent different amounts of damping
- higher damping gives a lower line with a broader peak at a lower frequency
- the peak of the highest line corresponds to F0, the natural frequency of the object
describe the effect of damping on resonance
As the amount of damping increases:
- Amplitude of vibration at any frequency decreases
- frequency of Max amplitude is lower
- peak becomes flatter and broader
describe how an MRI scanner works
- surrounding the scanner there are superconducting electromagnets and coils to produce radio waves
- These provide a very strong magnetic field
- hydrogen nuclei behave like tiny magnets and precess (sort of spin thing)
- when radio waves interact with them they resonate
- once the nuclei stop resonating they release the energy as different wavelength radio wave photons that are detected
Define SHM
an oscillatory motion where the acceleration of the oscillator is directly proportional to its displacement
and is directed towards a fixed / equilibrium point.
+ draw a graph
what does the defining SHM graph look like
a graph of a = -w^2x
so effectively y = -x but with gradient -w^2
what to remember to put when talking about the effect of amplitude changes on SHM
period is independent of amplitude
in what direction does the resultant force act on an object in SHM and why
towards the equilibrium position:
reason one:
- there are two forces acting, tension and weight, these cancel to make the horizontal force
reason two:
- the only acceleration occurs towards the equilibrium position therefore that is the direction of the resultant force
what can we derive from a v^2 against x^2 graph
v^2 = w^2A^2 - w^2x^2
so
y-int = (vmax)^2
grad = (angular frequency)^2
what does a kinetic energy against time graph look like for SHM (if starting from max displacement)
like a sin^2(x) graph because if
v = -sin(x) then
Ek = f((-sin^2(x))
it looks like a cos graph shifted up
what is the only thing that affects the period of a pendulum in SHM
the length of the pendulum
is the net force acting on a mass in a mass-spring system ever equal to its weight
ONLY when it passes through the equil point
what is the major difficulty in measuring angles of swing and period in an investigation and how can we solve it
- the angle of swing decreases with time due to the damping effect of air resistance
- to solve this you can measure a fraction of a swing (e.g. 1/4 of a swing) ACCURATELY using a DATA LOGGER and scale up to find period
where can resonance be a nuisance
- washing machines can have casings that resonate and cause loud vibrations
- wind/walkers can cause bridges to resonate
How to calculate the period of a simple pendulum when given length
T = 2(pi) sqrt(l/g)
