Chapter 17 Simple Harmonic Motion Flashcards
What is simple harmonic lotion basically compared to circular motion
What is oscisllting motion
Simple harmonic motion is a projection of circular motion onto the x axis
2) oscisllting motion is any motion thst REPEATS ITSELF
What is the difference between angular velocity and angular frequency?
Angular velocity is the change in theta by change in time
Whereas angular frequency is a CONSTANT = 2Pi/ T
- here angular frequency is used to describe periodicity .
- when circular motion, the thing is STILL PERIODICALLY moving in a circle, so angular frequency can be used to deccirbe there too
It just so happens to be that angular frewuencyn and velocity take the SAME VALUE in circular motion, both are constants
But in things not necessarily travelling in circles, but repeating (like ac current springs etc), the angular frequency described the motion if osccisltions
So are angular frequency and velocity related in any way
NO
Different meaning same symbol, just takes same value in circular motion
What is the definition of simple harmonic motion
Oscillation motion where the accelrwtion is directly PROPORTIONAL to the displacment and opposite the direction of displacment
Follows equation A =-x w^2
Remember SHM is just a projection of circular motion in x axis, looking at displacment
What do we approximate when doing SHM , big assumption
Known as free osccikstions ?
That there is no dampening force whatsoever ,
Assuming this it will oscisllte forever
Formulas for displacment of x
When do we use which one
2) how to replace theta as a function of time (wt)
X = Acos w t
X = A sin wt
We use when we define where we start from, kf start from max displacment = A , and angle to thr horoxntsl use cos
If starts from rest and angle to the vertical sin
2) if angular frequency is a constsnt , 2Pi/T then the angle = w t
Why is the time period independent if the amplitude
This because as amplitude increases, so does linear speed so time period would be the same
We saw if x starts at max or min displacement
What if in between?
This just a transformation of your graph by - Phase difference
So EQUATUIN becomes a = x cos( wt + phase difference)
Time period is a constsnt , due to angular frequency being a constsnt, and thus not depending on amplitude at all
What is the only thing that caused time period to change
How ti prove ?
It’s the force constant K and mass
Root k/m
Nothing else ,
Can prove this using a spring system
What is DEFINITION AND REQUIREMENT OF SHM again
Osciallting motion where acceleration is directly proportional to displacement
- and OPPOSITE direction of displacment
How to derrive advanced velocity equation
Differentiate fir velocity
Find out cos in terms of x and A using displacment
Use sin2 + cos2 to rearrange
Should get v = +- w sqaure root (A2-x2)
So max velocity occurs at displacment = 0
Thud w A
How to prove facts about max velocity is when x = 0 , max a when x = max etc
(Graphs)
Draw x = Acos graph
Now velocity is - sin so draw
And accelrwtion is - cos
This clearly shows the facts
Also that max acceleration occurred at max displacment but in OPPSOSITE direction, clearly fulfills SHM ruke
What happens in a thoufsk swing
Released from max Positive displacement
- thus accelrwtion is max negative
- accelerates until at 0 you are at max velocity ( negative) , but a is 0
- goes past and now displacment is negative , acceleration is positive
- this makes negative v increase to 0 at the max displacment , now acceleration is max positibe
- this increases v to max positbe at displacment 0, but acceleration now 0
Basically each time it goes , the velocity and acceleration same direction until middle, to where accelrwtion opposite and then once it hits 0, it goes back to being same direction until 0
But a and x always opposite
Similarly displacment and velocity same direction for half turns
Energy ideas of inter conversions
What is total energy always
No friction, energy in system must always add up to be constant
Here the kinetic energy is transferred into potential energy and vice verca
When speed is 0, ptoentisl is max and kinetic is 0
When v is max, kinetic max and potential 0 as it is at 0 displacment
These always add up
Using a spring system horizontal ( as vertical involved gpe )
Show how to find the equations of energy for potential and TOTAL energy
And thus Det dine shapes of graphs
Total energy will be when v = 0, and so all potential = 1/2kx2 when x = A so 1/2kA2
Thus this is a parabola at intercept 0, with max energy when x =A
- then, ke is the difference between total energy and potential at any time
So 1/2KA2 - 1/2kx2
Thus is clearly a negative parasol with intercept 1/2Ka2, where intercept is at d = A
Then line above shows how energy always constsnt
If the system will experience kinetic energy which ti died how can we always find max energy
2) why don’t we use this equation for max ptoentisl instead
= when all energy ke
Thud when v = max
= A omega
2) we don’t do max gpe all the time because it could be a combo of both gpe and elastic PE
Energy NEVER CROSSES NEGATIVE AXIS
Bevause it can’t be negative ?
What is dampening and what does it do
Dampening is when an external force works to REDUCE the amplitudes of the wave
And also eventually the time period so it ends up in rest
What is a free oscillations definition and what frequency does it go at (what is the f called)
What is a forced oscillation definition and what frequency does that go at
1) When a mechanical system is displaced from equilibrium and then allowed to oscillate without any EXTERNAL FORCES, so freely, this is free oscillations
It will oscisllte at a frequency = to its own frequency, known as its natural frequency, depending on other constants
2) if there is a DRIVING force that forces oscialltioms, then this is a FORCED OSCILLATION
- a forced oscillation will always go at the same FREQUENCY as the deriving frequency
What is resonance
What happens at resonance
Thus what will happen to the amplitude of a point at resonance
This is when the driving frequency matched the NATURAL FREQUENCY of the system, causing resonance to occur .
When resonance occurs, thr MAXIMUM energy transfer between driving force and system occurs
Thus at resonance, you see INCREASED amplitude to MAXIMUM AMPLITUDE (due to being at max energy) (amplitude = energy)
Explain resonance effects using the Barton’s pendulums
What’s the setup
- what determines the natural frequencies of the pendulums
- what happens initally when mass displaced
- what ends up happening to one of the pendulums
- why
You have numerous pendulums if different lengths attached to the SAME wire.
Also attached is a weight connected to a wire
- each wire has its own natural frequency. In this case it is PROPORTIONAL TO ITS OWN LENTGH
It follows that when you put thr mass under SHM, it drives all the other pendulums to move at its frequency.
- however the pendulum with the SAME LENTGH at it moves at an INCREASED AMPLITUDE compared to others
== this is because as same lentgh, has same natural frequency of the driving force, and so the driving force matched the natural frequency and RESONANCE OCCURS. the MAXIMUM energy transfer takes place, leading to the BIGGEST AMPLITUDE SEEN
So why does a wine glass break when sing at right frequency
This is because singing at natural frequency of the glass, driving force = natural frequency, resonance occurs, greatest energy transfer takes place, leading to maximum amplitude of the object
- this causes it TO BREAK , as no dampening
How did milenkum bridge could’ve fell
2) what did they install instead ti make it not fall
Would sway in the wind
And residents would tend to MATCH their step with the sway, creating a driving force that ended up matching natural frequency of the bridge leading ti resonance
Amplitude too big for support and could break
Instead fixed and installed DAMPENERS
What effect does dampening have on the amplitude and frequency of the oscillating object
Amplitude DECREASED
Frequency also decreases by a bit , not as much
What happens with each form of dampening, light heavy and critical
To the amplitude and frequency
Amplitude decreases each time increasingly for ALL FREQUENCIES
2) Max amplitude occurs at a LOWER FREQUENCY to its natural frequency each time too
3) overall graph shifts down and to the left with each dampening increasing effect mor
Again 3 effects if dampening
Lower amplitude for all frequencies
Max amplitude occurs at lower frequency then natural
Graph shifts down and to to the left
Examples of light heavy and very heavy dampening
Light = air, amplitudes decrease, but not really time period
Heavy = like water, amplitude decreases and time period pretty much too , still a few oscislltioms
Very heavy critical = just decreases to 0, doesn’t even do a wave , time period decreases (honey )
