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# Lesson 1

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```Lesson 1 - Oscillations
вЂў Harmonic Motion
Circular Motion
вЂў Simple Harmonic
Oscillators
вЂ“ Linear Horizontal/Vertical
Mass-Spring Systems
вЂў Energy of Simple
Harmonic Motion
Math Prereqs
d
dпЃ±
d
dпЃ±
sin пЃ± пЂЅ
co s пЃ±
cos пЃ± пЂЅ
пЂ­ sin пЃ±
2пЃ°
2пЃ°
пѓІ cos пЃ± d пЃ± пЂЅ пѓІ sin пЃ± d пЃ± пЂЅ
0
1
2пЃ°
0
0
2пЃ°
пѓІ cos
0
2
пЃ±dпЃ± пЂЅ
1
2пЃ°
2пЃ°
пѓІ sin
0
2
пЃ±dпЃ± пЂЅ
1
2
Identities
sin пЃ± пЂ« cos пЃ± пЂЅ 1
2
2
cos пЂЁ пЃ± п‚± пЃ¦ пЂ© пЂЅ cos пЃ± cos пЃ¦
cos пЃ± пЂ« cos пЃ¦ пЂЅ 2 cos
пЃ±пЂ«пЃ¦
sin пЃ± sin пЃ¦
sin
2
cos пЃ± пЂЅ
2
1
2
e
п‚± iпЃ±
пЂ«
1
cos 2 пЃ±
2
пЂЅ cos пЃ± п‚± i sin пЃ±
пЃ±пЂ­пЃ¦
2
Math Prereqs
пЂЅ " T im e A verag e "
1
T
пѓІ
T
0
Example:
1
1
пѓ¶
пѓ¦ 2пЃ° пѓ¶пѓ№
2 пѓ¦ 2пЃ°
2 пѓ¦ 2пЃ° пѓ¶
t пѓ· dt пЂЅ пѓІ пѓЄ пЂ« cos пѓ§ 2
t пѓ· пѓє dt пЂЅ
cos пѓ§
t пѓ· пЂЅ пѓІ cos пѓ§
T 0
T 0 пѓ«2 2
2
пѓЁ T пѓё
пѓЁ T пѓёпѓ»
пѓЁ T пѓё
T
T
Harmonic
Relation to circular motion
x пЂЅ A co s пЂЁ пЃ± пЂ« пЃ¦ пЂ© пЂЅ A co s пЂЁ пЃ· t пЂ« пЃ¦ пЂ©
пЃ·пЂЅ
2пЃ°
T
Horizontal mass-spring
пѓҐ F пЂЅ ma
Frictionless
Fs пЂЅ пЂ­ kx
HookeвЂ™s Law:
2
пЂ­ kx пЂЅ m block
2
d x
dt
2
пЂ«
k
m block
d x
dt
2
x пЂЅ0
Solutions to differential equations
вЂў Guess a solution
вЂў Plug the guess into the differential equation
вЂ“ You will have to take a derivative or two
вЂў Check to see if your solution works.
вЂў Determine if there are any restrictions (required
conditions).
вЂў If the guess works, your guess is a solution, but it
might not be the only one.
вЂў Look at your constants and evaluate them using
initial conditions or boundary conditions.
Our guess
x пЂЅ A co s пЂЁ пЃ· t пЂ« пЃ¦ пЂ©
Definitions
x пЂЅ A co s пЂЁ пЃ· t пЂ« пЃ¦ пЂ©
вЂў Amplitude - (A) Maximum value of the displacement (radius of
circular motion). Determined by initial displacement and velocity.
вЂў Angular Frequency (Velocity) - пЂЁпЃ·пЂ© Time rate of change
of the phase.
вЂў Period - (T) Time for a particle/system to complete one cycle.
вЂў Frequency - (f) The number of cycles or oscillations completed in
a period of time
вЂў Phase - пЂЁпЃ·t пЂ« пЃ¦пЂ© Time varying argument of the trigonometric
function.
вЂў Phase Constant - пЂЁпЃ¦пЂ© Initial value of the phase. Determined by
initial displacement and velocity.
The restriction on the solution
k
пЃ· пЂЅ
2
m block
f пЂЅ
T пЂЅ
пЃ·
2пЃ°
2пЃ°
пЃ·
1
k
2пЃ°
m block
пЂЅ 2пЃ°
m block
пЂЅ
k
The constant вЂ“ phase angle
x пЂЁt пЂЅ 0пЂ© пЂЅ A
пЃ¦пЂЅ0
v пЂЁt пЂЅ 0пЂ© пЂЅ 0
v пЂЅ пЂ­ A пЃ· sin пЂЁ пЃ· t пЂ« пЃ¦ пЂ©
x пЂЅ A co s пЂЁ пЃ· t пЂ« пЃ¦ пЂ©
a пЂЅ пЂ­ A пЃ· co s пЂЁ пЃ· t пЂ« пЃ¦ пЂ©
2
x пЂЁt пЂЅ 0пЂ© пЂЅ 0
v пЂЁ t пЂЅ 0пЂ© пЂЅ v0
пЃ¦пЂЅ
пЃ°
2
Energy in the SHO
E пЂЅ
1
mv пЂ«
2
2
vпЂЅп‚±
1
kx пЂЅ
2
2
k
m
1
2
пЂЁA пЂ­ x
2
2
kA
2
Average Energy in the SHO
x пЂЅ A co s пЂЁ пЃ· t пЂ« пЃ¦ пЂ©
U пЂЅ
1
k x
2
пЂЅ
2
dx
dt
K пЂЅ
2
m v
2
kA
2
cos
2
vпЂЅ
1
1
пЂЅ
1
2
1
kA
2
4
пЂЅ пЂ­ A пЃ· sin пЂЁ пЃ· t пЂ« пЃ¦ пЂ©
mпЃ· A
2
2
2
sin
2
K пЂЅ U
1
4
mпЃ· A пЂЅ
2
2
1
4
kA
2
Example
вЂў A mass of 200 grams is connected to a light spring that has
a spring constant (k) of 5.0 N/m and is free to oscillate on a
horizontal, frictionless surface. If the mass is displaced 5.0
cm from the rest position and released from rest find:
вЂў a) the period of its motion,
вЂў b) the maximum speed and
вЂў c) the maximum acceleration of the mass.
вЂў d) the total energy
вЂў e) the average kinetic energy
вЂў f) the average potential energy
Damped Oscillations
вЂњDashpotвЂќ
Fdam ping пЂЅ пЂ­ bv
пЂ­ kx пЂ­ b
dx
пЂЅ ma
dt
2
Equation of Motion
Solution
m
d x
dt
2
x пЂЅ Ae
пЂ«b
dx
пЂ« kx пЂЅ 0
dt
пЂ­ пЃЎt
cos пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ©
x пЂЅ Ae
vпЂЅ
dx
пЂ­ пЃЎt
пЂЅ пЂ­ Ae
пЂ­ пЃЎt
пЂЅ пЂ­Ae
пЂ­ пЃЎt
dt
cos пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ©
пЃ· п‚ў sin пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ© пЂ« A пЂЁ пЂ­ пЃЎ пЂ© e
aпЂЅ
dt
2
пЂЅ пЂ­ Ae
пЂЅ Ae
2
d x
dt
2
Ae
пЂ­ пЃЎt
пЂ«
пЂ­ пЃЎt
пЂ­ пЃЎt
cos пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ©
пѓ©пѓ« пЃ· п‚ў sin пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ© пЂ« пЃЎ cos пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ© пѓ№пѓ»
2
d x
пЂ­ пЃЎt
пЃ· п‚ў cos пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ© пЂ« A пЃЎ e
2
пЃ» 2 пЃЎ пЃ· п‚ў sin пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ© пЂ« пѓ©пѓ« пЃЎ
b dx
k
пЂ«
m dt
2
пЂ­ пЃЎt
пЃ· п‚ў sin пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ© пЂ« A пЃЎ e
пЂ­ пЃЎt
пЃ· п‚ў sin пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ© пЂ« A пЃЎ e
2
пЂ­ пЃЎt
cos пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ©
пЃЅ
2
пЂ­ пЃ· п‚ў пѓ№пѓ» cos пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ©
xпЂЅ0
m
пЃ» 2 пЃЎ пЃ· п‚ў sin пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ© пЂ« пѓ©пѓ« пЃЎ
пЂ­
Ae
пЃЎ пЂЅ
b
t
2m
2
пЃЅ
b
k
2
пЂ­ пЃЎt
пЂ­ пЃЎt
пЂ­ пЃ· п‚ў пѓ№пѓ» cos пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ© пЂ­
Ae
A e cos пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ© пЂЅ 0
пѓ©пѓ« пЃ· п‚ў sin пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ© пЂ« пЃЎ cos пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ© пѓ№пѓ» пЂ«
m
m
b
пѓ№
пЃ· п‚ў пѓє s in пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ© пЂ«
пѓ­ пѓЄ 2 пЃЎ пЃ·п‚ў пЂ­
m пѓ»
пѓ®пѓ«
b
2m
b
k пѓ№
пѓј
п‚ў2
п‚ў
пѓЄ пЃЎ пЂ­ пЃ· пЂ­ m пЃЎ пЂ« m пѓє cos пЂЁ пЃ· t пЂ« пЃ¦ пЂ© пѓЅ пЂЅ 0
пѓ«
пѓ»
пѓѕ
2
пѓ¦ b пѓ¶
2
пЂ­пѓ§
пѓ· пЂ­ пЃ·п‚ў пЂЅ 0
m пѓЁ 2m пѓё
k
пЃ·п‚ў пЂЅ
пѓ¦ b пѓ¶
пЂ­пѓ§
пѓ·
m пѓЁ 2m пѓё
k
2
Damped frequency oscillation
пЃЎ пЂЅ
b
2m
k
пЃ·п‚ў пЂЅ
m
пЂ­
b
2
4m
2
b п‚і 4mk
2
B - Critical damping (=)
C - Over damped (>)
Giancoli 14-55
вЂў A 750 g block oscillates on the end of a spring
whose force constant is k = 56.0 N/m. The mass
moves in a fluid which offers a resistive force F =
-bv where b = 0.162 N-s/m.
вЂ“ What is the period of the motion? What if there had
been no damping?
вЂ“ What is the fractional decrease in amplitude per cycle?
вЂ“ Write the displacement as a function of time if at t = 0,
x = 0; and at t = 1.00 s, x = 0.120 m.
Forced vibrations
Fext пЂЅ F0 cos пЃ· t
пЂ­ kx пЂ­ b
dt
2
m
dx
d x
dt
2
пЂ«b
dx
dt
пЂ« F0 cos пЃ· t пЂЅ m a
пЂ« kx пЂЅ F0 cos пЃ· t
x пЂЅ A 0 sin пЂЁ пЃ· t пЂ« пЃ¦ 0 пЂ©
Resonance
x пЂЅ A 0 sin пЂЁ пЃ· t пЂ« пЃ¦ 0 пЂ©
k
пЃ·0 пЂЅ
Natural frequency
m
F0
A0 пЂЅ
m
пЂЁпЃ·
2
пЂ­пЃ·
2
0
2
b пЃ·
2
пЂ«
m
2
пѓ¦ m пЂЁ пЃ· 2 пЂ­ пЃ· 02 пЂ© пѓ¶
пЂ­1
пѓ·
пЃ¦ 0 пЂЅ tan пѓ§
пѓ§
пѓ·
bпЃ·
пѓЁ
пѓё
2
Quality (Q) value
вЂў Q describes the sharpness of
the resonance peak
вЂў Low damping give a large Q
вЂў High damping gives a small Q
вЂў Q is inversely related to the
fraction width of the resonance
peak at the half max amplitude
point.
Q пЂЅ
mпЃ·0
b
пЃ„пЃ·
пЃ·0
пЂЅ
1
Q
пЃ„пЃ·
Tacoma Narrows Bridge
Tacoma Narrows Bridge (short clip)
```
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