DEVIL PHYSICS
THE BADDEST CLASS ON CAMPUS
IB PHYSICS
TOPIC 9
WAVE PHENOMENA
TSOKOS LESSON 9-1
SIMPLE HARMONIC MOTION
Essential Idea:
 The solution of the harmonic oscillator can
be framed around the variation of kinetic
and potential energy in the system.
Nature Of Science:
 Insights:
 The equation for simple harmonic motion (SHM)
can be solved analytically and numerically.
 Physicists use such solutions to help them to
visualize the behavior of the oscillator.
 The use of the equations is very powerful as any
oscillation can be described in terms of a
combination of harmonic oscillators.
 Numerical modeling of oscillators is important in
the design of electrical circuits.
Understandings:
 The defining equation of SHM
 Energy changes
Applications And Skills:
 Solving problems involving acceleration,
velocity and displacement during simple
harmonic motion, both graphically and
algebraically
 Describing the interchange of kinetic and
potential energy during simple harmonic
motion
 Solving problems involving energy transfer
during simple harmonic motion, both
graphically and algebraically
Guidance:
 Contexts for this sub-topic include the
simple pendulum and a mass-spring
system.
Data Booklet Reference:
2

T
2
a  x
x  x0 sin t

1
E K  m 2 xo2  x 2
2
1
ET  m 2 xo2
2
x  x0 cos t
l
Pendulum : T  2
g
v  x0 cos t
v  x0 sin t
v  
x
2
o
 x2


m
Mass  spring : T  2
k
Utilization:
 Fourier analysis allows us to describe all
periodic oscillations in terms of simple
harmonic oscillators. The mathematics of
simple harmonic motion is crucial to any
areas of science and technology where
oscillations occur
 The interchange of energies in oscillation
is important in electrical phenomena
Utilization:
 Quadratic functions (see Mathematics HL
sub-topic 2.6; Mathematics SL sub-topic
2.4; Mathematical studies SL sub-topic
6.3)
 Trigonometric functions (see Mathematics
SL sub-topic 3.4)
Aims:
 Aim 4: students can use this topic to
develop their ability to synthesize complex
and diverse scientific information
 Aim 7: the observation of simple harmonic
motion and the variables affected can be
easily followed in computer simulations
Aims:
 Aim 6: experiments could include (but are not
limited to): investigation of simple or torsional
pendulums; measuring the vibrations of a tuning
fork; further extensions of the experiments
conducted in sub-topic 4.1. By using the force law,
a student can, with iteration, determine the
behaviour of an object under simple harmonic
motion. The iterative approach (numerical
solution), with given initial conditions, applies
basic uniform acceleration equations in successive
small time increments. At each increment, final
values become the following initial conditions.
Oscillation vs. Simple
Harmonic Motion
 An oscillation is any motion in which the
displacement of a particle from a fixed
point keeps changing direction and there
is a periodicity in the motion i.e. the
motion repeats in some way.
 In simple harmonic motion, the
displacement from an equilibrium
position and the force/acceleration are
proportional and opposite to each other.
Relating SHM to Motion
Around A Circle
Radians
 One radian is defined as
the angle subtended by
an arc whose length is
equal to the radius
l

r
lr
 1
Radians
Circumference  2r
l

r
l  2r
Circumference  2 rad 
360  2 rad 
Angular Velocity


t
l
v
t
l
  , l  r
r

vr
 r
t
Angular
Acceleration

a
t
2
v
ar 
r
v  r

r 
a
2
r
 r
2
Data Booklet Reference:
2

T
2
a  x
x  x0 sin t

1
E K  m 2 xo2  x 2
2
1
ET  m 2 xo2
2
x  x0 cos t
l
Pendulum : T  2
g
v  x0 cos t
v  x0 sin t
v  
x
2
o
 x2


m
Mass  spring : T  2
k
Period
2r
vT 
T
vT  r
2

T
2
T 

Data Booklet Reference:
2

T
2
a  x
x  x0 sin t

1
E K  m 2 xo2  x 2
2
1
ET  m 2 xo2
2
x  x0 cos t
l
Pendulum : T  2
g
v  x0 cos t
v  x0 sin t
v  
x
2
o
 x2


m
Mass  spring : T  2
k
Frequency
T
2

1
f 
T

f 
2
  2f
Relating SHM to Motion
Around A Circle
 The period in one complete oscillation of
simple harmonic motion can be likened to the
period of one complete revolution of a circle.
angle swept
Time taken = ---------------------angular speed (ω)
T
2

2

T
Mass-Spring System
Relating SHM to Motion
Around A Circle, Mass-Spring
F  ma
 kx  ma
k
a x
m
2
a   r
k
 r x
m
k
2
 
m
k

m
2
Relating SHM to Motion
Around A Circle, Mass-Spring
2

T 

k
m
2

k
m
T  2
m
k
Data Booklet Reference:
2

T
2
a  x
x  x0 sin t

1
E K  m 2 xo2  x 2
2
1
ET  m 2 xo2
2
x  x0 cos t
l
Pendulum : T  2
g
v  x0 cos t
v  x0 sin t
v  
x
2
o
 x2


m
Mass  spring : T  2
k
Relating SHM to Motion
Around A Circle, Pendulum
ma   mg sin 
a   g sin 
θ
FT
mg cosθ
mg sinθ
mg
Displaceme nt
x  L
x

L
Relating SHM to Motion
Around A Circle, Pendulum
θ
FT
mg cosθ
mg sinθ
mg
a   g sin 
x

L
x
a   g sin
L
Small
x x
sin   
L L
Relating SHM to Motion
Around A Circle, Pendulum
a   g sin 
θ
FT
mg cosθ
mg sinθ
mg
x x
sin   
L L
x
g
a  g   x
L
L
2
a   x
g
 
L
2
Relating SHM to Motion
Around A Circle, Pendulum
g
 
L
2
2
T


g
L
2
θ
FT
mg cosθ
mg sinθ
mg
T  2
L
g
Data Booklet Reference:
2

T
2
a  x
x  x0 sin t

1
E K  m 2 xo2  x 2
2
1
ET  m 2 xo2
2
x  x0 cos t
l
Pendulum : T  2
g
v  x0 cos t
v  x0 sin t
v  
x
2
o
 x2


m
Mass  spring : T  2
k
Relating SHM to Motion
Around A Circle, Pendulum
a   x
2
θ
2
FT
mg cosθ
mg sinθ
mg
d x
a 2
dt
2
d x
2
  x
2
dt
2
d x
2
 x  0
2
dt
Relating SHM to Motion
Around A Circle, Pendulum
d 2x
2
 x  0
2
dt
θ
FT
mg cosθ
mg sinθ
mg
Differential
Equation
Machine
x  x0 cost 
x  x0 sin t 
Data Booklet Reference:
2

T
2
a  x
x  x0 sin t

1
E K  m 2 xo2  x 2
2
1
ET  m 2 xo2
2
x  x0 cos t
l
Pendulum : T  2
g
v  x0 cos t
v  x0 sin t
v  
x
2
o
 x2


m
Mass  spring : T  2
k
Relating SHM to Motion
Around A Circle, Pendulum
x  x0 cost 
θ
FT
Max Displacement
mg cosθ
mg sinθ
mg
x  x0 sin t 
Zero Displacement
Relating SHM to Motion
Around A Circle, Pendulum
x  x0 cost 
x  x0 sin t 
θ
FT
mg cosθ
mg sinθ
mg
dx
v
 x0 cost 
dt
dx
v
 x0 sin t 
dt
Data Booklet Reference:
2

T
2
a  x
x  x0 sin t

1
E K  m 2 xo2  x 2
2
1
ET  m 2 xo2
2
x  x0 cos t
l
Pendulum : T  2
g
v  x0 cos t
v  x0 sin t
v  
x
2
o
 x2


m
Mass  spring : T  2
k
Relating SHM to Motion
Around A Circle, Pendulum
x  x0 cost 
x  x0 sin t 
θ
FT
mg cosθ
mg sinθ
mg
So what is x0?
dx
v
 x0 cost 
dt
dx
v
 x0 sin t 
dt
Relating SHM to Motion
Around A Circle, Pendulum
x  x0 cost 
x  x0 sin t 
θ
FT
mg cosθ
mg sinθ
mg
So what is x0?
Amplitude
dx
v
 x0 cost 
dt
dx
v
 x0 sin t 
dt
Equation Summary
 Defining Equation:
a   x
 Angular Frequency:
2

 2f
T
 Period:
T
2

2
Equation Summary
x  x0 cost 
v  x0 sin t 
a   x0 cost 
2
x  x0 sin t 
v  x0 cost 
a   x0 sin t 
2
Equation Summary
 Maximum speed:
v0  x0
 Maximum acceleration:
a   x0
2
Data Booklet Reference:
2

T
2
a  x
x  x0 sin t

1
E K  m 2 xo2  x 2
2
1
ET  m 2 xo2
2
x  x0 cos t
l
Pendulum : T  2
g
v  x0 cos t
v  x0 sin t
v  
x
2
o
 x2


m
Mass  spring : T  2
k
Energy in SHM
1
2
E K  mv
2
v  x0 sin t 
1
2
2
2
E K  m x0 sin t 
2
Energy in SHM
1
2
2
2
E K  m x0 sin t 
2
vmax  x0
E K max
1
2
2
 m x0
2
Energy in SHM
1
2
2
E K max  m x0
2
ET  E K max  E P  E K
E P  ET  E K
1
1
2
2
2
2
2
E P  m x0  m x0 sin t 
2
2
Energy in SHM
1
1
2
2
2
2
2
EP  m x0  m x0 sin t 
2
2
1
2
2
2
EP  m x0 1  sin t 
2
1
2
2
2
EP  m x0 cos t 
2




Energy in SHM
 

1
2
2
2
E P  m x0 cos t 
2
x  x0 cost 
x  x0 cos t 
2
2
2
1
2 2
E P  m x
2
Energy in SHM
v  x0 sin t 
v   x0 sin t 
2
2
2
2


v   x0 1  cos t 
2
2
2
2

x  x 

v   x0   x0 cos t 
2
v 
2
2
2
2
2
2
2
0
v   x0  x
2
2
2
2
Data Booklet Reference:
2

T
2
a  x
x  x0 sin t

1
E K  m 2 xo2  x 2
2
1
ET  m 2 xo2
2
x  x0 cos t
l
Pendulum : T  2
g
v  x0 cos t
v  x0 sin t
v  
x
2
o
 x2


m
Mass  spring : T  2
k
Energy in SHM
1
2
E K  mv
2
v   x0  x 2
2

1
2
2
2
E K  m x0  x
2

Data Booklet Reference:
2

T
2
a  x
x  x0 sin t

1
E K  m 2 xo2  x 2
2
1
ET  m 2 xo2
2
x  x0 cos t
l
Pendulum : T  2
g
v  x0 cos t
v  x0 sin t
v  
x
2
o
 x2


m
Mass  spring : T  2
k
End Result
Section Summary:
x  x0 cost 
1
2v 2 x sin 
t
0
E K max 2 m x0
x2 1x0 sin2 
t 2
 21
22
2
2 E  m
2 x  x  2
E

mv

v   x0K sin

t
o
T
K


2dx
T

2
E

E

E

E
 Ta  2 Kx max
P
K
v2 
 2x0 cos
t 
2
2
1
t
2 2 t 
dt
2
2 x

v

1
cos
E

m

xo
0
v



x

x
T
1
2
0
E Px 
E

E
2
2
2
T
K
x
sin

t
l 0
dx
1
E P 2 2 m 2 x0 2cos 2
t 
2


v




x
sin

t
v
f



v


x


x
cos

t
0
2
1
2
l
0
0
2
2
x Etx1
t2m2 x01  xPendulum
2dt : T2  2
2

0K cos
T


E P  m
x

m

x
sin

t
x 2 x0 cos
2 0 t2 
0 2
g
2
l
v 2 x0  x
2
v


x
cos

t
0

  , l  r
2
2
2
x  x0 2 cos t f  m
r
2
v  x0 sin t v   Mass
 spring
: T  2
x

x
2

k
0
1
 2
2 2
2
E P  m x   2f
v v r  xr


x
o

t


2



Essential Idea:
 The solution of the harmonic oscillator can
be framed around the variation of kinetic
and potential energy in the system.
Understandings:
 The defining equation of SHM
 Energy changes
Applications And Skills:
 Solving problems involving acceleration,
velocity and displacement during simple
harmonic motion, both graphically and
algebraically
 Describing the interchange of kinetic and
potential energy during simple harmonic
motion
 Solving problems involving energy transfer
during simple harmonic motion, both
graphically and algebraically
Data Booklet Reference:
2

T
a  2 x
x  x0 sin t; x  x0 cos t
v  x0 cos t; x  x0 sin t
v  
x
2
o
 x2 
1
E K  m 2 xo2  x 2 
2
1
E K  m 2 xo2
2
l
Pendulum : T  2
g
m
Mass  spring : T  2
k
Aims:
 Aim 4: students can use this topic to
develop their ability to synthesize complex
and diverse scientific information
 Aim 7: the observation of simple harmonic
motion and the variables affected can be
easily followed in computer simulations
Aims:
 Aim 6: experiments could include (but are not
limited to): investigation of simple or torsional
pendulums; measuring the vibrations of a tuning
fork; further extensions of the experiments
conducted in sub-topic 4.1. By using the force law,
a student can, with iteration, determine the
behaviour of an object under simple harmonic
motion. The iterative approach (numerical
solution), with given initial conditions, applies
basic uniform acceleration equations in successive
small time increments. At each increment, final
values become the following initial conditions.
Homework
#1-13