Chapter 17 Simple Harmonic Motion

Question 1

What is simple harmonic lotion basically compared to circular motion

What is oscisllting motion

Answer 1

Simple harmonic motion is a projection of circular motion onto the x axis

2) oscisllting motion is any motion thst REPEATS ITSELF

Question 2

What is the difference between angular velocity and angular frequency?

Answer 2

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

Question 3

So are angular frequency and velocity related in any way

Answer 3

NO

Different meaning same symbol, just takes same value in circular motion

Question 4

What is the definition of simple harmonic motion

Answer 4

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

Question 5

What do we approximate when doing SHM , big assumption
Known as free osccikstions ?

Answer 5

That there is no dampening force whatsoever ,

Assuming this it will oscisllte forever

Question 6

Formulas for displacment of x

When do we use which one

2) how to replace theta as a function of time (wt)

Answer 6

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

Question 7

Why is the time period independent if the amplitude

Answer 7

This because as amplitude increases, so does linear speed so time period would be the same

Question 8

We saw if x starts at max or min displacement
What if in between?

Answer 8

This just a transformation of your graph by - Phase difference

So EQUATUIN becomes a = x cos( wt + phase difference)

Question 9

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 ?

Answer 9

It’s the force constant K and mass

Root k/m

Nothing else ,

Can prove this using a spring system

Question 10

What is DEFINITION AND REQUIREMENT OF SHM again

Answer 10

Osciallting motion where acceleration is directly proportional to displacement
- and OPPOSITE direction of displacment

Question 11

How to derrive advanced velocity equation

Answer 11

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

Question 12

How to prove facts about max velocity is when x = 0 , max a when x = max etc

(Graphs)

Answer 12

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

Question 13

What happens in a thoufsk swing

Answer 13

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

Question 14

Energy ideas of inter conversions

What is total energy always

Answer 14

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

Question 15

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

Answer 15

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

Question 16

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

Answer 16

= 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

Question 17

Energy NEVER CROSSES NEGATIVE AXIS

Answer 17

Bevause it can’t be negative ?

Question 18

What is dampening and what does it do

Answer 18

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

Question 19

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

Answer 19

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

Question 20

What is resonance

What happens at resonance

Thus what will happen to the amplitude of a point at resonance

Answer 20

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)

Question 21

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

Answer 21

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

Question 22

So why does a wine glass break when sing at right frequency

Answer 22

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

Question 23

How did milenkum bridge could’ve fell

2) what did they install instead ti make it not fall

Answer 23

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

Question 24

What effect does dampening have on the amplitude and frequency of the oscillating object

Answer 24

Amplitude DECREASED

Frequency also decreases by a bit , not as much

Question 25

What happens with each form of dampening, light heavy and critical

To the amplitude and frequency

Answer 25

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

Question 26

Again 3 effects if dampening

Answer 26

Lower amplitude for all frequencies

Max amplitude occurs at lower frequency then natural

Graph shifts down and to to the left

Question 27

Examples of light heavy and very heavy dampening

Answer 27

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 )