Chalter 17 SHM INCOMPLETE Flashcards
What is the difference between angular velocity and angular frequency?
Angular velocity describes the change in angle with the change of time
Angular frequency however refers to anything that has OSCILLATING phenomena ! This could involve anything that repeats, such as a pendulum, ac current etc.
- confusion is why same symbol. Here it is possible that in some cases the va,he for angular frequency and velocity is THE SAME.
This is because in something like circular motion which is REPEATING in a circle, can describe with angular frequency, but also as an angle is subtended angular velocity
Some cases no angle is subtended however, thus only angular frequency if it repeats
Again angular frequency and velocity related in any way?
NO angular velocity is a function of time whereas angular frequency is a CONSTANT
When is angular frequency same as angular velocity
What is angular velocity defined as and what is angular frequency always defined as
This is only in uniform circular motion
As angular velocity now is ALSO A CONSTANTZ!
2) angular velocity is change in theta by change in time, yh when you doing one complete revelation it gonna be the same
But angular frequency is always 2Pi/ at or 2PiF
What does simple harmonic motion mea
Where the periodicity is sinusoids cosine repeated curved
Simple, harmonic motion is from free oscillations? What does this mean
No driving force to make the oscillation and no friction force to top pose,
We approximating no frictionnhere
- this is WITHIUT DAMPING
In SHM what is max displacement called
Amplitude
What are the formulas for displacement in x axis
When use
X = Acos(wt)
X = Asin(wt)
Use cos if it starts from max displacement =A
And sin if it starts from equilibrium
As cos 0 is 1
And sin 0 is 9
But what if it starts anywhere on the x axis, what is equation then
This is a transformation by thr phase difference
So cos or sin (wt + phase difference) simoly
What is the definition and conditions for SHM
Oscillating motion where the acceleration of the object is
- DIRECTLY PROPORTIONAL to the displacement from equilibrium
- Always OPPOSITE the direction of displacement
What’s going on in this definition for SHM
When an object experiencing SHM is displaced from equilibrium, a restoring force is felt such that an acceleration created so that it returns back to equilibrium .
As a result, the acceleration at equilibrium is 0 as its there, and further out the more you want it to accelerate back
Thus the acceleration is proportional to displacement from centre and opposite in its direction so that it always faces the centr
How to write this proportionality rule for SHM for acceleration
A=-kx
A=-w^2x
Is the acceleration of an object in SHM dependent on its amp,iTunes of swing or what?
No solely dependent on angular frequency that’s it, if the amplitude increase so will speed hence
How to find equations for different systems for what various things would be
Find thr restorimg firce using physics ideas and find acceleration equation with f=ma
Now eauate this acceleration to -w2x and then ewuate coefficients !
Can then rearrange a for frequency or time period and show what variables it’s equal to
How to derive displacement horizontal
Assuming max displacement so A, x = Acos x, but as we need x function of time we know w= x/t, so x=wt (not proper but meh)
So x=Acos(wt)
And if it starts at rest
X=Asin(wt)
As cos 0 when t = 0 = 1 = A
And sin 0 when t is 0 = 0 which matches up
What about if it doesn’t start at max displacement or at equilibrium temperature
Need a phase shift constant to move graph
So X= Acos (wt + alpha)
Okay how about acceleration derivation
Need the second derrivative of x respect to t
Need chain rule for this, and idea that derrivative of sin is cos and cos is -sin
Eventually you see you can sub in equation for x into Accelrtaion
Giving you A= -w2x
Which is the definition and requirement for simple harmonic motion
What about advanced velocity equation
Advanced is that with you velocity equation frim differentiating first , you can use trig identify sin2 + cos2 to rearrange, then sub in using equation from dispatch,centripetal and rearrange
What is the advanced velocity equation
V= w (square root of A2-x2)
Using velocity equation how do you know when vmax is
We know vmax is when it’s at centre so displacement is so so it becomes
Vmax = WA
Now explain how we know when exactly max a is at and max velocity
Max velocity is when gradient of displacement time graph is greatest. This is actually 0 when it’s at max displac,ent , and maximum at equilibrium.
Thus we know that at max displacements it’s min velocity and equilibrium = max velocity
2) similarly acceleration is gradient of velocity time graph
Again we know this is maximum when velocity is 0, which is at max displacement
And minimum when velocity is max, which is when eaukibkrum
This matches the a = -w2x , as a is proportional to x but opposite it’s direction
So what happen in a typical swing simple harmonic
Typical swing from max displac ent acceleration is max here velocity isn0 but accelerates to the centr gaining speed but losing acceleration until max speed at centre but no acceleration.
Then last this it decelerates essentially so speed decreases until zero , and acceleration magnitude increases until at extreme ,and then it accelerated so that speed increases until at centre it’s max again and acceleration isn0
Properijslmto displacment!
So what directions are velocity in seen as though acceleration and displcemnt always opposite each other
Velocity is in same direction as acceleration for a bit until equilibrium where acceleration is opppote direction whereas velocity still in that druecutonnuntik zero and then velocity joins back ti acclertaion drecitoj but only until equilibrium again
As a result velocity and acceleration are in same and opposite directions!
Velocity and displacement direcions?
X
Energy ideas now, in a pendulum what energies are there and assuming energy constant where are the energies maximum
Gpe and ke are being interconverted
When v is 0 at extreme so is ke and this is max height so all energy in gpe
But when height is min at equilibrium which is also vmax , all energy kinetic is Et= 1/2mv2 where v is vmax
Ideas of max kinetic and max POTETNIAL is the same for any system . Thus how can we calculate max energy if any system knowing there will always be kinetic
Etot = 1/2mv2 where v is v max = AW
So 1/2 m (Aw)2!
How to determine the shape of ke and potential energy against displacement
What plane in a mass spring system should you use to explain
Using a mass spring system in the horizontal plane allows potential energy to only be elastic and not gravatuaonsk if it was vertical
We know that epe is e tot at extreme displacment , so 1/2kx2 becomes 1/2kA2 and that’s e tot
As we want to express ke in terms of horixntosl displacment keep it in terms of epe.
Ke at any point = ETot - epe
Epe at any point is jus 1/2kx2
So ke = 1/2KA2- 1/2kx2
And first is constant, so this is an INVERSE shape quadratic because of minus
And so at equilibrium where it’s max ke as max v ke = 1/2Ka2 which we know is etot so mark that, when x =A (min ke ) this gin cancel to give 0 so mark that
Also y intercept is 1/2ka2 = e tot
Similarly we said potential energy at any point= 1/2kx2, just constants along with positive quadratic
We know epe greatest at max displacment based in equation so when x =A , gives max energy and lowest when x=0 so no energy
Finally as constants in both is 1/2k shape is same
Finally what’s is constant throughout all energy graphs
Total energy, should add up
Also why will energy graphs never cross x axis?
I because energy a scalar can’t take these values! COMES UP IN BARE MULTIPLE CHOICE
And so what’s main energy equation takeaway
Total energy = 1/2mvmax 2 where vmax is aw
As when v= vmax all energy is kinetic
As kinda abed to find same equation for when all energy is potential form
Another way ti think about why it’s quadratic curve for ke
Ke properinsl to v2
V peiptinsk to x as equaruon square roots x
So ke is promotional to x 2
Giving a quadratic
DAMINPIMF AND RESONANCE
How to tell if exponential decay
Decrease by same factor each time
What is dampening
This is when an external force actually works to decrease the amplitude and periods of the oscillations
Examples of light, heavy and very heavy dampening
Light dampening reduces social lotions but barely affects the time period, an expanse of this is air resistance with a pendulum
Heavy dampening will reduce both, like a pendulum in water
Very heavy dampening almost doesn’t even allow oscillations , kills completely like lil
Natural frequency and forced / free social lotions?
Free oscillations are without external driving factors causing it to socialite at a frequency
If a driving factor causes it to socialite at a frequency = to its own natural frequency it will resonate, where amplitude increases a lot
Natural frequency is the frequency it vibrates at when it is at free oscillations so
Resonance
Occurs when the driver force resonance is = to natural frequency of object
This increases amplitude considerably and is the greatest possible energy transfer
Here it can cause object like wine glass to break
Same thing happened with Tacoma narrows bridge?
Resonance caused by the wind caused it to collapse
Here millennium bridge, dampeners were needed to be installed to make sure this would happened either when resistance was achieved
What does time period depend on in shm only
Only depend on k and m
Not amplitude
Can prove this using a spring system
Quickly how to prove time period only depends on force CONSTSNT and mass in any system?
Use spring
Force restoring = -Kx by hooked law = ma of system
SHM , so a = -w2x, f = -mw2x
Thus -mw2= -k
And omega is thus Döner on k and m only
So timer period indoednet on everything elde
What damping returns the body to rest the FASTEST?
Thid will be CRITICAL DAMPENING (2ND OPTION)
Although heavy dampening is larger external force, deceleration causes it to return to “ to long , but only in one oscillation
2nd option returns to 0 the fastest but with a few more oscillations
What would be the aim in designing something to stop for the time taken to come to rest snd the oscislltioms
You want time taken to be quick and the oscislltiosn to be less do it’s smooth
What happens to TIME PERIODS specifically for all 3 dampening as we know amplitudes gonna decrease
1) basically constant
2) increases a bit
3) increases more
Why does the amolitude decrease in dampening simple answer
Damoenign is when external force reduces amolitudes, so energy transferred energy OUT OF THE SYSTEM into heat, reducing amplotiud as energy decreases e
What do 3 dampening s do to smolotude but more importnently frewuency at which MAX AMPLITUDE OCCURS
Reduced amplitudes , oscillations
2) the frewuency at which max amplitude was at devreases to the left Esch damoenign
So overall graoh goes down and left
So if it was critical vs heavy dampening for a amplitude against frequency graph of a driving force, what would it look like
Critical = FREQUENCY of resonance Peak amplitude will shift to left
- all amplitudes will shift downwards l
- the first few cycles are MAINTAINED so osccisltions
Heavy = frequency shift even more left
- all amplitudes shift even more down such thst it just GOES TO EQUILIBRIUM IN INE GO!
osccisltions are removed
Remmeber how are heavy and critical dampening differnet
Heavy will return to EQUILBKRUM straight away never goes BELOW THE X ACIS, but it takes longer to do this
Critical does have a few oscislltioms but returns much FASTER
