Chalter 17 SHM INCOMPLETE

Question 1

What is the difference between angular velocity and angular frequency?

Answer 1

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

Question 2

Again angular frequency and velocity related in any way?

Answer 2

NO angular velocity is a function of time whereas angular frequency is a CONSTANT

Question 3

When is angular frequency same as angular velocity

What is angular velocity defined as and what is angular frequency always defined as

Answer 3

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

Question 4

What does simple harmonic motion mea

Answer 4

Where the periodicity is sinusoids cosine repeated curved

Question 5

Simple, harmonic motion is from free oscillations? What does this mean

Answer 5

No driving force to make the oscillation and no friction force to top pose,

We approximating no frictionnhere

• this is WITHIUT DAMPING

Question 6

In SHM what is max displacement called

Answer 6

Amplitude

Question 7

What are the formulas for displacement in x axis
When use

Answer 7

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

Question 8

But what if it starts anywhere on the x axis, what is equation then

Answer 8

This is a transformation by thr phase difference

So cos or sin (wt + phase difference) simoly

Question 9

What is the definition and conditions for SHM

Answer 9

Oscillating motion where the acceleration of the object is
- DIRECTLY PROPORTIONAL to the displacement from equilibrium
- Always OPPOSITE the direction of displacement

Question 10

What’s going on in this definition for SHM

Answer 10

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

Question 11

How to write this proportionality rule for SHM for acceleration

Answer 11

A=-kx
A=-w^2x

Question 12

Is the acceleration of an object in SHM dependent on its amp,iTunes of swing or what?

Answer 12

No solely dependent on angular frequency that’s it, if the amplitude increase so will speed hence

Question 13

How to find equations for different systems for what various things would be

Answer 13

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

Question 14

How to derive displacement horizontal

Answer 14

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

Question 15

What about if it doesn’t start at max displacement or at equilibrium temperature

Answer 15

Need a phase shift constant to move graph

So X= Acos (wt + alpha)

Question 16

Okay how about acceleration derivation

Answer 16

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

Question 17

What about advanced velocity equation

Answer 17

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

Question 18

What is the advanced velocity equation

Answer 18

V= w (square root of A2-x2)

Question 19

Using velocity equation how do you know when vmax is

Answer 19

We know vmax is when it’s at centre so displacement is so so it becomes
Vmax = WA

Question 20

Now explain how we know when exactly max a is at and max velocity

Answer 20

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

Question 21

So what happen in a typical swing simple harmonic

Answer 21

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!

Question 22

So what directions are velocity in seen as though acceleration and displcemnt always opposite each other

Answer 22

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!

Question 23

Velocity and displacement direcions?

Answer 23

X

Question 24

Energy ideas now, in a pendulum what energies are there and assuming energy constant where are the energies maximum

Answer 24

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

Question 25

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

Answer 25

Etot = 1/2mv2 where v is v max = AW
So 1/2 m (Aw)2!

Question 26

How to determine the shape of ke and potential energy against displacement

What plane in a mass spring system should you use to explain

Answer 26

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

Question 27

Finally what’s is constant throughout all energy graphs

Answer 27

Total energy, should add up

Question 28

Also why will energy graphs never cross x axis?

Answer 28

I because energy a scalar can’t take these values! COMES UP IN BARE MULTIPLE CHOICE

Question 29

And so what’s main energy equation takeaway

Answer 29

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

Question 30

Another way ti think about why it’s quadratic curve for ke

Answer 30

Ke properinsl to v2
V peiptinsk to x as equaruon square roots x
So ke is promotional to x 2

Giving a quadratic

Question 31

DAMINPIMF AND RESONANCE

Question 32

How to tell if exponential decay

Answer 32

Decrease by same factor each time

Question 33

What is dampening

Answer 33

This is when an external force actually works to decrease the amplitude and periods of the oscillations

Question 34

Examples of light, heavy and very heavy dampening

Answer 34

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

Question 35

Natural frequency and forced / free social lotions?

Answer 35

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

Question 36

Resonance

Answer 36

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

Question 37

Same thing happened with Tacoma narrows bridge?

Answer 37

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

Question 38

What does time period depend on in shm only

Answer 38

Only depend on k and m

Not amplitude

Can prove this using a spring system

Question 39

Quickly how to prove time period only depends on force CONSTSNT and mass in any system?

Use spring

Answer 39

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

Question 40

What damping returns the body to rest the FASTEST?

Answer 40

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

Question 41

What would be the aim in designing something to stop for the time taken to come to rest snd the oscislltioms

Answer 41

You want time taken to be quick and the oscislltiosn to be less do it’s smooth

Question 42

What happens to TIME PERIODS specifically for all 3 dampening as we know amplitudes gonna decrease

Answer 42

1) basically constant
2) increases a bit
3) increases more

Question 43

Why does the amolitude decrease in dampening simple answer

Answer 43

Damoenign is when external force reduces amolitudes, so energy transferred energy OUT OF THE SYSTEM into heat, reducing amplotiud as energy decreases e

Question 44

What do 3 dampening s do to smolotude but more importnently frewuency at which MAX AMPLITUDE OCCURS

Answer 44

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

Question 45

So if it was critical vs heavy dampening for a amplitude against frequency graph of a driving force, what would it look like

Answer 45

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

Question 46

Remmeber how are heavy and critical dampening differnet

Answer 46

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