chapter 15 - oscillationschapter 15 - oscillations simple harmonic oscillator (sho) energy in sho...
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![Page 1: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/1.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Chapter 15 - Oscillations
“The pendulum of the mind oscillates between sense andnonsense, not between right and wrong.”
-Carl Gustav Jung
David J. StarlingPenn State Hazleton
PHYS 211
![Page 2: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/2.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Simple Harmonic Oscillator (SHO)
Oscillatory motion is motion that is periodic in
time (e.g., earthquake shakes, guitar strings).
The period T measures the time for one oscillation.
![Page 3: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/3.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Simple Harmonic Oscillator (SHO)
Oscillatory motion is motion that is periodic in
time (e.g., earthquake shakes, guitar strings).
The period T measures the time for one oscillation.
![Page 4: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/4.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Simple Harmonic Oscillator (SHO)
Oscillatory motion that is sinusoidal is known as
Simple Harmonic Motion.
x(t) = xm cos(ωt)
xm is the maximum displacementω is the angular frequency: ω = 2πf = 2π/T .
![Page 5: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/5.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Simple Harmonic Oscillator (SHO)
Oscillatory motion that is sinusoidal is known as
Simple Harmonic Motion.
x(t) = xm cos(ωt)
xm is the maximum displacement
ω is the angular frequency: ω = 2πf = 2π/T .
![Page 6: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/6.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Simple Harmonic Oscillator (SHO)
Oscillatory motion that is sinusoidal is known as
Simple Harmonic Motion.
x(t) = xm cos(ωt)
xm is the maximum displacementω is the angular frequency: ω = 2πf = 2π/T .
![Page 7: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/7.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Simple Harmonic Oscillator (SHO)
Two oscillators can have different frequencies, or
different phases:
x(t) = xm cos(ωt + φ)
![Page 8: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/8.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Simple Harmonic Oscillator (SHO)
Sinusoidal oscillations are described by these
definitions:
x(t) = xm cos(ωt + φ)
T = 1/f
ω = 2πf
I x in meters
I T in seconds
I f in Hertz (1/s)
I ω in rad/s
![Page 9: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/9.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Simple Harmonic Oscillator (SHO)
Sinusoidal oscillations are described by these
definitions:
x(t) = xm cos(ωt + φ)
T = 1/f
ω = 2πf
I x in meters
I T in seconds
I f in Hertz (1/s)
I ω in rad/s
![Page 10: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/10.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Simple Harmonic Oscillator (SHO)
Since we know the position of an oscillating
object, we also know its velocity and acceleration:
x(t) = xm cos(ωt + φ)
v(t) =dxdt
= −ωxm sin(ωt + φ)
a(t) =dvdt
= −ω2xm cos(ωt + φ)
![Page 11: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/11.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Simple Harmonic Oscillator (SHO)
The maximum velocity and acceleration depend
on the frequency and the maximum displacement.
Max position: xm
Max velocity: ωxm
Max acceleration: ω2xm
![Page 12: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/12.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Simple Harmonic Oscillator (SHO)
Simple harmonic motion is generated by a linear
restoring force:
F = −kx = ma = md2xdt2
d2xdt2 +
km
x = 0
The solution to this differential equation:
x(t) = xm cos(√
k/mt + φ)
(so ω =√
k/m)
![Page 13: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/13.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Simple Harmonic Oscillator (SHO)
Simple harmonic motion is generated by a linear
restoring force:
F = −kx = ma = md2xdt2
d2xdt2 +
km
x = 0
The solution to this differential equation:
x(t) = xm cos(√
k/mt + φ)
(so ω =√
k/m)
![Page 14: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/14.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Simple Harmonic Oscillator (SHO)
Simple harmonic motion is generated by a linear
restoring force:
F = −kx = ma = md2xdt2
d2xdt2 +
km
x = 0
The solution to this differential equation:
x(t) = xm cos(√
k/mt + φ)
(so ω =√
k/m)
![Page 15: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/15.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Simple Harmonic Oscillator (SHO)
The motion of a SHO is related to motion in a
circle.
x(t) = xm cos(ωt + φ)
v(t) = −ωxm sin(ωt + φ)
a(t) = −ω2xm cos(ωt + φ)
![Page 16: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/16.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Simple Harmonic Oscillator (SHO)
The motion of a SHO is related to motion in a
circle.
x(t) = xm cos(ωt + φ)
v(t) = −ωxm sin(ωt + φ)
a(t) = −ω2xm cos(ωt + φ)
![Page 17: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/17.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Simple Harmonic Oscillator (SHO)
The motion of a SHO is related to motion in a
circle.
x(t) = xm cos(ωt + φ)
v(t) = −ωxm sin(ωt + φ)
a(t) = −ω2xm cos(ωt + φ)
![Page 18: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/18.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Chapter 15
Lecture Question 15.1The graph below represents the oscillatory motion of threedifferent springs with identical masses attached to each.Which of these springs has the smallest spring constant?
(a) Graph 1
(b) Graph 2
(c) Graph 3
(d) Both 2 and 3 are smallest and equal
(e) All three have the same spring constant.
![Page 19: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/19.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Energy in SHO
A spring stores potential energy. To find it,
calculate the work the spring force does:
∆U = −W = −∫ x2
x1
(−kx)dx =12
kx22 −
12
kx21
A spring compressed by x stores energy U = 12 kx2.
![Page 20: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/20.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Energy in SHO
A spring stores potential energy. To find it,
calculate the work the spring force does:
∆U = −W = −∫ x2
x1
(−kx)dx =12
kx22 −
12
kx21
A spring compressed by x stores energy U = 12 kx2.
![Page 21: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/21.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Energy in SHO
As a mass oscillates, the energy transfers from
kinetic to potential energy.
At the ends of the motion, velocity is zero, K is zero and Uis maximum.
![Page 22: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/22.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Energy in SHO
As a mass oscillates, the energy transfers from
kinetic to potential energy.
At the ends of the motion, velocity is zero, K is zero and Uis maximum.
![Page 23: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/23.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Energy in SHO
The energy oscillates between:
U =12
kx2 =12
kx2m cos2(ωt + φ)
K =12
mv2 =12
mω2︸︷︷︸k
x2m sin2(ωt + φ)
E = U + K = U =12
kx2m
![Page 24: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/24.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Energy in SHO
The energy oscillates between:
U =12
kx2 =12
kx2m cos2(ωt + φ)
K =12
mv2 =12
mω2︸︷︷︸k
x2m sin2(ωt + φ)
E = U + K = U =12
kx2m
![Page 25: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/25.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Energy in SHO
The energy oscillates between:
U =12
kx2 =12
kx2m cos2(ωt + φ)
K =12
mv2 =12
mω2︸︷︷︸k
x2m sin2(ωt + φ)
E = U + K = U =12
kx2m
![Page 26: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/26.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Pendulums
A simple pendulum is a ball on a string. It acts
like a SHO for small angles.
The restoring force:
F = −mg sin θ ≈ −mgθ
I = mL2
![Page 27: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/27.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Pendulums
For small angles (θ < 20◦), a pendulum is like a
simple harmonic oscillator.
τ = −(mgθ)L = Iα = Id2θ
dt2
d2θ
dt2 +mgL
I︸︷︷︸ω2
θ = 0
θ(t) = θm cos(ωt + φ)
ω =√
mgL/I =√
g/L
T = 1/f = 2π/ω = 2π√
L/g
![Page 28: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/28.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Pendulums
For small angles (θ < 20◦), a pendulum is like a
simple harmonic oscillator.
τ = −(mgθ)L = Iα = Id2θ
dt2
d2θ
dt2 +mgL
I︸︷︷︸ω2
θ = 0
θ(t) = θm cos(ωt + φ)
ω =√
mgL/I =√
g/L
T = 1/f = 2π/ω = 2π√
L/g
![Page 29: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/29.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Pendulums
For small angles (θ < 20◦), a pendulum is like a
simple harmonic oscillator.
τ = −(mgθ)L = Iα = Id2θ
dt2
d2θ
dt2 +mgL
I︸︷︷︸ω2
θ = 0
θ(t) = θm cos(ωt + φ)
ω =√
mgL/I =√
g/L
T = 1/f = 2π/ω = 2π√
L/g
![Page 30: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/30.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Pendulums
For small angles (θ < 20◦), a pendulum is like a
simple harmonic oscillator.
τ = −(mgθ)L = Iα = Id2θ
dt2
d2θ
dt2 +mgL
I︸︷︷︸ω2
θ = 0
θ(t) = θm cos(ωt + φ)
ω =√
mgL/I =√
g/L
T = 1/f = 2π/ω = 2π√
L/g
![Page 31: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/31.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Pendulums
For small angles (θ < 20◦), a pendulum is like a
simple harmonic oscillator.
τ = −(mgθ)L = Iα = Id2θ
dt2
d2θ
dt2 +mgL
I︸︷︷︸ω2
θ = 0
θ(t) = θm cos(ωt + φ)
ω =√
mgL/I =√
g/L
T = 1/f = 2π/ω = 2π√
L/g
![Page 32: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/32.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Pendulums
For a physical pendulum with moment of inertia I
and small oscillations,
d2θ
dt2 +mgh
Iθ = 0
ω =√
mgh/I
T = 1/f = 2π/ω = 2π√
I/mgh
![Page 33: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/33.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Pendulums
For a physical pendulum with moment of inertia I
and small oscillations,
d2θ
dt2 +mgh
Iθ = 0
ω =√
mgh/I
T = 1/f = 2π/ω = 2π√
I/mgh
![Page 34: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/34.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Pendulums
For a physical pendulum with moment of inertia I
and small oscillations,
d2θ
dt2 +mgh
Iθ = 0
ω =√
mgh/I
T = 1/f = 2π/ω = 2π√
I/mgh
![Page 35: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/35.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Pendulums
For a physical pendulum with moment of inertia I
and small oscillations,
d2θ
dt2 +mgh
Iθ = 0
ω =√
mgh/I
T = 1/f = 2π/ω = 2π√
I/mgh
![Page 36: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/36.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Pendulums
A torsion pendulum is a symmetric object where
the restoring torque arises from a twisted wire.
τ = −κθ (similar to spring)
d2θ
dt2 +κ
Iθ = 0
ω =√κ/I
T = 1/f = 2π/ω = 2π√
I/κ
![Page 37: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/37.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Pendulums
A torsion pendulum is a symmetric object where
the restoring torque arises from a twisted wire.
τ = −κθ (similar to spring)d2θ
dt2 +κ
Iθ = 0
ω =√κ/I
T = 1/f = 2π/ω = 2π√
I/κ
![Page 38: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/38.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Pendulums
A torsion pendulum is a symmetric object where
the restoring torque arises from a twisted wire.
τ = −κθ (similar to spring)d2θ
dt2 +κ
Iθ = 0
ω =√κ/I
T = 1/f = 2π/ω = 2π√
I/κ
![Page 39: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/39.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Pendulums
A torsion pendulum is a symmetric object where
the restoring torque arises from a twisted wire.
τ = −κθ (similar to spring)d2θ
dt2 +κ
Iθ = 0
ω =√κ/I
T = 1/f = 2π/ω = 2π√
I/κ
![Page 40: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/40.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Chapter 15
Lecture Question 15.3A grandfather clock, which uses a pendulum to keepaccurate time, is adjusted at sea level. The clock is thentaken to an altitude of several kilometers. How will theclock behave in its new location?
(a) The clock will run slow.
(b) The clock will run fast.
(c) The clock will run the same as it did at sea level.
(d) The clock cannot run at such high altitudes.
![Page 41: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/41.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Damped Oscillations
Damped simple harmonic motion is the result of
oscillatory behavior in the presence of a retarding
force.
Fd = −bv
with b the damping constant.
![Page 42: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/42.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Damped Oscillations
Applying Newton’s Second Law to this situation,
Fnet = ma
−kx− bv = ma
0 =d2xdt2 +
bm
dxdt
+km
x
The solution:
x(t) = xme−bt/2m cos(ω′t + φ)
ω′ =
√km− b2
4m2
![Page 43: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/43.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Damped Oscillations
Applying Newton’s Second Law to this situation,
Fnet = ma
−kx− bv = ma
0 =d2xdt2 +
bm
dxdt
+km
x
The solution:
x(t) = xme−bt/2m cos(ω′t + φ)
ω′ =
√km− b2
4m2
![Page 44: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/44.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Damped Oscillations
For damped harmonic motion, the oscillations
will die out over time.
Side note: ω′ =√
km −
b2
4m2 is only valid if k/m > b2/4m2.
What happens if k/m ≤ b2/4m2?
![Page 45: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/45.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Damped Oscillations
For damped harmonic motion, the oscillations
will die out over time.
Side note: ω′ =√
km −
b2
4m2 is only valid if k/m > b2/4m2.
What happens if k/m ≤ b2/4m2?
![Page 46: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/46.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Damped Oscillations
If an oscillator of angular frequency ω is drivenby an external force at a frequency ωd, then theresponse will also be at ωd.
d2xdt2 +
km
x = F0 cos(ωt)→ x(t) = A cos(ωt + φ)
The amplitude A depends on the relationship between ωd
and ω.
![Page 47: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/47.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Damped Oscillations
If an oscillator of angular frequency ω is drivenby an external force at a frequency ωd, then theresponse will also be at ωd.
d2xdt2 +
km
x = F0 cos(ωt)→ x(t) = A cos(ωt + φ)
The amplitude A depends on the relationship between ωd
and ω.
![Page 48: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/48.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Damped Oscillations
If an oscillator of angular frequency ω is drivenby an external force at a frequency ωd, then theresponse will also be at ωd.
d2xdt2 +
km
x = F0 cos(ωt)→ x(t) = A cos(ωt + φ)
The amplitude A depends on the relationship between ωd
and ω.
![Page 49: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/49.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Damped Oscillations
If a damped oscillator is driven at its naturalfrequency, the system is on resonance and theoscillations are maximum.
The width of the resonance peak depends on the dampingconstant.
![Page 50: Chapter 15 - OscillationsChapter 15 - Oscillations Simple Harmonic Oscillator (SHO) Energy in SHO Pendulums Damped Oscillations Chapter 15 Lecture Question 15.1 The graph below represents](https://reader030.vdocument.in/reader030/viewer/2022040104/5e56a56623694947a11600ff/html5/thumbnails/50.jpg)
Chapter 15 - Oscillations
Simple HarmonicOscillator (SHO)
Energy in SHO
Pendulums
Damped Oscillations
Damped Oscillations
If a damped oscillator is driven at its naturalfrequency, the system is on resonance and theoscillations are maximum.
The width of the resonance peak depends on the dampingconstant.