Unit 3.2 Vibrations Flashcards
1
Q
Definition of Simple Harmonic Motion (SHM)?
(5-way…. wow)
A
- If a body, subject to a restoring force,
- moves in such a way that acceleration
- is directed towards a fixed point in its path
- and is ∝ to its displacement from that point
- the object is moving with SHM
2
Q
Define isochronous
A
At equal time intervals
(Shoutout to Galileo Galilei)
3
Q
What does it mean if it’s oscillating?
(3-way… understand it)
A
- Object performing certain motion repeatedly
- w/ time,
- undergoing a periodic motion
4
Q
Anyway, what’s the equation that models SHM?
(In data booklet)
A
a = -ω2x
5
Q
Define ω2
(SHM equation)
A
Constant of proportionality
6
Q
Define ω?
(SHM equation)
A
Angular velocity/frequency (rads-1)
(Pretty much relates to a … topic…)
7
Q
2 things to show that an oscillating system is obeying SHM?
(1 additional “NB”)
A
- Acceleration ∝ to its displacement
- Acceleration always directed towards centre of oscillation
(Some systems only approximate SHM; very small oscillations for pendulum)
8
Q
Define amplitude
(A)
A
Point of max displacement from equilibrium position
(m)
9
Q
Define displacement
(x)
A
How far object is from equilibrium position
(m)
10
Q
Define period
(T)
A
Time for 1 complete oscillation
(s)
11
Q
Define frequency
(F?.. lol)
A
N° of oscillations per sec
(Hz OR s-1)
12
Q
Define angular frequency (ω)
(3-way… also is velocity ig)
A
- The rate of change of
- angular displacement
- w/ respect to time
(rads1)
13
Q
The equation for frequency?
(In AS data booklet but re-arranged)
A
f = 1/T
14
Q
2 equations to gaining angular frequency?
(Only 1 in data booklet but re-arranged)
(ONE OF THEM NOT IN DATA BOOKLET… can be kinda derived easily)
A
- ω = 2π/T
- ω = 2πf
15
Q
You must learn the sine and cos curves
Both degrees & radians… actually why don’t u draw it?
A
Bummer
(To the specific page…. at least it converges right…? ¬.¬)
16
Q
How would u make the sine curve turn into a cos curve?
Draw it B|
A
let y = sin (θ + π/2)
…. bruh
17
Q
Remember differentiating displacement, velocity, maybe acceleration (lol no)
And also, what happens if u differentiate sin or cos… I’ll list them here
(Truly a money add then multiply….)
A
- Differentiation (Displacement -> Velocity -> Acceleration)
2…
- let y = xn, dy/dx = nxn-1
- let y = sin nx, dy/dx = ncos(nx)
- let y = cos nx, dy/dx = -nsin(nx)
- let y = ncos nx, dy/dx = -n2sin(nx)
- let y = nsin nx, dy/dx = n2cos(nx)
18
Q
How to show relationship between circular motion and SHM?
(I’ll only show 1 equation…. which is also in the data booklet)
(Otherwise still end equation also not in data booklet)
A
DRAW IT, WTFFFFF???
ω = θ/t
change to θ = ωt
…
end result is x = Acosωt
19
Q
Since there’s a relation between circular motion and SHM, how do we derive an expression for velocity w/ respect to time?
(2 things)
A
- so, x = Acosωt
- dx/dt = -ωAsinωt = v
20
Q
WHICH THEN, the origin of a = -ω2(x) from gaining expression for velocity FROM relation between circular motion and SHM???
A
- V = -ωAsinωt
- dv/dt = -ω2Acosωt
- HENCE, a = -ω2x
OMGGG
21
Q
For graphing SHM, where does sin graph start?
(1,1)
A
At equilibrium position, when t & x = 0
22
Q
Show the 2 equations that starts at equilibrium?
A
- x = Asinωt
- V = ωAcosωt
23
Q
For graphic SHM, where does cos graph start?
(1,1)
A
At max displacement position, when t = 0 & x = A
24
Q
Show the 2 equations that start at top of oscillation?
A
- x = Acosωt
- v = -ωAsinωt
25
I mean then u must know of already know what a sin and cos graph look like?
(Draw it)
Hence, draw the negative version of a sin and cos graph
(Ig important for these different expressions of x, v & a)
You got it boss.
Okay boss.
26
What's the graph to prove SHM?
(5-way)
- Acceleration versus displacement
- a ∝ -x
- (draw it)
- gradient = -ω2
- a = **-ω2**x
27
What can we gain from velocity versus displacement?
(Draw it... and it's **not in the data booklet**)
- You got it bro
- THE SPECIAL EQUATION:
v2 = ω2 (A2 - x2)
28
What's the equation u use to find max acceleration?
max a = ω2A
29
What's the equation u use to find max velocity?
max v = ωA
30
Tell me about the phase angle aspect?
(4 things)
- We know sin starts form equilibrium
- And cos starts from max amplitude
- Technically cos is a sin but ig a few seconds later (iykyk)
- Hence, phase difference of 90° or π/2 radians
31
THEREFORE, the phase angle equation?
(Major check-up)
x = Asin(ωt + Ɛ)
(Ɛ = phase angle)
(starts between equilibrium at top???)
32
Tell me about the energy in an oscillator?
(1 + 2-way)
- Total energy always remains constant
- Due to the conservation of energy
- in a closed system
33
Draw the graph of **energy versus displacement**
(Also rando formulaes?)
Got it BOSS
There's also these formulae (u apparently don't needa know) but meh:
1. Total Energy at any position = ½mω2A2
**Total Energy = KE + PE at any displacement**
2. PE = ½mω2x2
when x = A, PEmax = ½mω2A2
3. KE = ½mω2(A2 - x2)
when x=0, KEmax = ½mω2A2
34
But then, tell me about the energy when any system oscillates?
(Energy versus time)
Continual exchange between PE and KE
35
Show the form taken for potential energy stored in an oscillator
(3 step by steps)
(... perhaps figure out where i can start from)
1. PE ∝ x2
2. also x = Asinωt
3. PE ∝ sin2(ωt)
36
Show the form taken for kinetic energy stored in an oscillator
(3 step by steps)
(... likewise to previous)
1. KE ∝ v2
2. v = Aωcosωt
3. KE ∝ cos2(ωt)
37
Energies stored in an oscillator... draw graph of energy vs time between PE and KE
(1 tip + 2 points)
I got it boss
- Dotted straight line hitting the peaks,
and know these 2 points:
- Always positive
- Total energy is conserved
38
What can we change F = ma to due to linkage between circular motion and SHM?
(Grasped it?)
F = mω2x
(Remember a = ω2A in which A is pretty much distance)
39
What's the equation for the period of a mass on a spring?
(In data booklet)
T = 2π√m/k
40
What 3 equations do we need to derive T = 2π√m/k?
1. F = kx (Hooke's law)
2. F = mω2x
3. ω = 2π/T
41
I want you to derive T = 2π√m/k
(Tips be like)
O_o
- Remember about the reciprocal:
2π/√m/k -> 2π√m/k = T
- Or you can just swap sides for each but root stays same
42
What's the equation for the period of a pendulum?
(In data booklet)
(ALSO only for small angles 5°/less sinθ = θ
T = 2π√l/g
43
What 4 equations needed to derive T = 2π√l/g
(**Each has brief desc.**)
(1 thing needs a check up)
1. F = mω2x (From previous)
2. F = mgsinθ (Will explain later)
3. θ = x/l (Also dunno where it comes from)
4. ω = 2π/T (From previous)
44
How is even F = mgsinθ derived?
(Deriving T = 2π√l/g)
(4 step by steps)
- w = mg so F = mg
- At an angle ∴ splits into components
- Weight is downwards ∴ looking for vertical component
- Voila: F = mgsinθ
45
I also want you to derive T = 2π√l/g
(Tips again?)
- Remember about reciprocal lol:
2π/√g/l -> 2π√l/g = T
- Likewise, you can just swap sides also wary of root
46
Describe forced oscillations?
(3 things)
- External periodic force applied to a system
- Rate of input energy = rate of dissipation
- Has no damping forces (friction from air)
47
"Properties" of forced oscillations?
(2-way + 2-way)
- Amplitude of "forced oscillations" greatest when
- driving freq. matches **natural freq.** (f0) of system
- If system forced to oscillate at its natural freq.
- , it is **resonating**
48
So then what does it mean if it's at resonance?
(This is literally the simple ver. of "properties)
(2-way)
- Driving freq. = natural freq. (f0)
- Amplitude will be at it's greatest
49
Can you draw the graph of resonance of forced oscillations
All yours
50
Ofc, also what's an example of forced oscillations?
Child pushed on swing at regular intervals
51
Brief description of Barton's pendulum?
(2-way)
- "Driver pendulum"
- w/ several attached pendulums in different lengths
52
What's the practical example of forced oscillations?
Barton's pendulum
53
How does Barton's pendulum showcase forced oscillation?
(3 things)
- As driver pendulum set into motion
- Each attached pendulum undergoes forced oscillation
- Others have low amplitude unless...
54
How does Barton's pendulum showcase resonance?
(4-way)
- 1 pendulum same length as driver
- and also has largest (max) amplitude likewise to driver
- as it has the same f0 as driver
- hence, they're in phase (is what it seems to me)
55
Define damping?
(2-way)
- Gradual loss of energy in oscillating system
- due to resistive forces
56
Describe the effect of damping on resonance?
(1 thing + 2-way)
- Increased damping reduces amplitude of oscillations
- Increasing damping slightly reduces
- the freq. w/ maximum amplitude
57
Draw the graph describing the effect of damping on resonance?
In summary:
- No damping = high amplitude
- Light to Medium damping = lower amplitudes
- Heavy damping = lowest amplitude
Also there's that resonant frequency which I believe is also the natural frequency
58
What are 4 examples of resonance then?
(1 of them can't provide info D:)
(I'll try bold what's imp)
1. Tuned circuits
2. Radio circuits
3. Microwave cooking
4. Tacoma Narrows Bridge
59
Explain resonance of radio circuits?
(3-way)
- Radio tuned by **setting freq.** equal to radio station
- Electric circuit **resonates at same freq.** as specific broadcast
- Resonance allows **signal to be amplified**
60
Explain resonance of microwave cooking?
(2-way + 1)
- Microwaves at **particular freq. can resonate**
- w/ **H2O molecules** in food
- Vibrations = **friction between molecules** = produce heat
61
Explain resonance of Tacoma Narrows bridge?
(1 + 2-way)
- Bridge can **vibrate in wind/earthquakes**
- As bridge vibrates **at resonant freq.**
- may **break** due to being **amplified**
62
Anyway, describe free oscillations?
(2-way)
- Un-damped
- Amplitude remains constant
63
Describe damped oscillations?
(2-way)
- Light damping
- Amplitude decays exponentially w/ time
64
So then, if we would increase damping, what may just happen?
(Literally we've stated this at a previous card yano?)
(3-way)
- May alter freq. of oscillating system
- Period may increase slightly
- Causes freq. to decrease
65
What are 3 main types of damping?
1. Light damping (Under damping)
2. Heavy damping (Over damping)
3. Critical damping
66
Describe light damping?
+ draw graph :v
(2 points)
- Decreasing amplitude of oscillation
- Constant time period for each oscillation
67
Describe heavy damping?
+ draw graph ,':)
(3 points)
- Amplitude decreases slowly
- Returns to equilibrium after long period of time
- No oscillations
68
Describe critical damping?
+ draw graph :b
(2-way)
- Returns to equilibrium
- in shortest possible time
69
Well in order that I stated, tell me damping in terms of **analogy of a swing**?
1. Swing decreases naturally
2. Stops swing quickly but under control
3. Stops swing straight away (emergency stop)
70
HHHEEEENCE, 3 examples of damping?
(Will also try bold imp parts too)
1. Car suspension damping
2. Door return
3. Piano string damping
71
Explain car suspension damping?
(2 points)
- For comfortable ride, need heavy damping
- To prevent constant bounce/stopping suddenly
72
Explain door return?
(2 points)
- Closes door automatically...
- Ensures door moves at safe controlled rate
73
Explain piano string damping?
(1 thing + 2-way)
- Stops notes ringing after letting go....
- Overridden using sustain pedal
- which stops felt dampers returning to string
74
Nice, but well, remember all in
:D
