phy 102: waves & quanta topic 1 oscillations john cockburn (j.cockburn@... room e15)
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PHY 102: Waves & Quanta Topic 1 Oscillations John Cockburn (j.cockburn@... Room E15). Simple Harmonic Motion revision Displacement, velocity and acceleration in SHM Energy in SHM Damped harmonic motion Forced Oscillations Resonance. Simple Harmonic Motion. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: PHY 102: Waves & Quanta Topic 1 Oscillations John Cockburn (j.cockburn@... Room E15)](https://reader036.vdocument.in/reader036/viewer/2022062323/5681582d550346895dc592a7/html5/thumbnails/1.jpg)
PHY 102: Waves & Quanta
Topic 1
Oscillations
John Cockburn (j.cockburn@... Room E15)
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•Simple Harmonic Motion revision
•Displacement, velocity and acceleration in SHM
•Energy in SHM
•Damped harmonic motion
•Forced Oscillations
•Resonance
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Simple Harmonic Motion
Object (eg mass on a spring, pendulum bob) experiences restoring force F, directed towards the equilibrium position, proportional in magnitude to the displacement from equilibrium:
m
k
xdt
xd
kxdt
xdm
kxF
22
2
2
2
Solution:
x(t) =
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Example: Simple Pendulum (small amplitude)
(NOT SHM for large amplitude)
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SHM and Circular motion
Light
Displacement of oscillating object = projection on x-axis of object undergoing circular motion
y(t) = Acos
For rotational motion with angular frequency , displacement at time t:
y(t) = Acos(t + )
= angular displacement at t=0 (phase constant)A = amplitude of oscillation (= radius of circle)
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Velocity and acceleration in SHM
)( :onAccelerati
)( :Velocity
)cos()( :ntDisplaceme
2
2
dt
xdta
dt
dxtv
tAtx
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Velocity and acceleration in SHM
)cos()( :onAccelerati
)sin( )( :Velocity
)cos()( :ntDisplaceme
22
2
tAdt
xdta
tAdt
dxtv
tAtx
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Energy in Simple Harmonic Motion
Potential energy:
Work done to stretch spring by an amount dx = Fdx = -kxdx
Total work done to stretch spring to displacement x:
energy potential storedW
So, at any time t, the potential energy of the oscillator is given by:
PE
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Energy in Simple Harmonic Motion
Kinetic Energy:
At any time t, kinetic energy given by:
22
2
1
2
1
dt
dxmmvKE
E
Total Energy at time t = KE + PE:
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Energy in Simple Harmonic Motion
Conclusions
Total energy in SHM is constant
222
2
1
2
1kAAmE
Throughout oscillation, KE continually being transformed into PE and vice versa, but TOTAL ENERGY remains constant
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Energy in Simple Harmonic Motion
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Damped Oscillations
In most “real life” situations, oscillations are always damped (air, fluid resistance etc)
In this case, amplitude of oscillation is not constant, but decays with time..(www.scar.utoronto.ca/~pat/fun/JAVA/dho/dho.html)
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Damped Oscillations
For damped oscillations, simplest case is when the damping force is proportional to the velocity of the oscillating object
In this case, amplitude decays exponentially:
)'cos()( )2( tAetx tmb
Equation of motion: dt
dxbkx
dt
xdm
2
2
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Damped Oscillations
NB: in addition to time dependent amplitude, the damped oscillator also has modified frequency:
2
22
02
2
44'
m
b
m
b
m
k
Light Damping(small b/m)
Heavy Damping(large b/m)
Critical Damping
2
2
4m
b
m
k
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Forced Oscillations & Resonance
If we apply a periodically varying driving force to an oscillator (rather than just leaving it to vibrate on its own) we have a FORCED OSCILLATION
Free Oscillation with damping:
02
2
kxdt
dxb
dt
xdm
tmbeA )2(0 Amplitude
Forced Oscillation with damping:
tFkxdt
dxb
dt
xdm Dcos02
2
2
22
02
2
44'
m
b
m
b
m
k Frequency
2D
222D
0
m-k
F Amplitude
b
MAXIMUM AMPLITUDE WHEN DENOMINATOR MINIMISED:k = mD
2 ie when driving frequency = natural frequency of the UNDAMPED oscillator
m
kD 0
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Forced Oscillations & Resonance
When driving frequency = natural frequency of oscillator, amplitude is maximum.
We say the system is in RESONANCE
“Sharpness” of resonance peak described by quality factor (Q)
High Q = sharp resonance
Damping reduces Q
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The Millennium Bridge