Advanced Higher Physics Unit 1 Simple Harmonic Motion (SHM)
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Transcript of Advanced Higher Physics Unit 1 Simple Harmonic Motion (SHM)
Advanced Higher Physics Unit 1
Simple Harmonic Motion(SHM)
OscillationAn oscillation is a regular to and fro movement around a zero point.For example, a simple pendulum or a mass on a spring (horizontally or Vertically).
zero point
zero point
Examples of SHM
For each example of SHM, state how Ep is stored and where Ek is coming from:
Example Ep stored as: Ek coming from:
Pendulum
Mass on a spring
Trolley moving between springs
Cork floating
Potential energy of mass
Elastic energy of spring
Elastic energy of springs
Potential energy of cork
Moving mass
Moving mass
Moving trolley
Moving cork
Period and frequency
Tw
2
An oscillation has a:
•period (T) which is the time taken for one complete cycle
•Frequency (f) which is the number of cycles made per second.
fw 2
Not in data booklet. In data booklet.
fT
1
In data booklet(higher part)
Link between SHM and circular motion
Circular motion SHM
turntable shadow
Restoring force
For all oscillations there is an unbalanced force acting on the object, and this force is opposite to the motion – so it is called a restoring force.
If this restoring force is proportional to the displacement from thezero position then the oscillation is called Simple Harmonic Motion.
Fun=0N
F
F
weight
tension
Zero position
Simple Harmonic motion
kya
ya
yF
2
2
dt
yda
Where y is the displacement from the zero position
Since F=ma and m is constant.
However as
ywdt
yd 22
2
and k is defined as w² for SHM
In data booklet
ym
k
dt
yd
kydt
ydmmaF
2
2
2
2
The constant is related to the period of the motion by m
km
kw 2
Therefore ywdt
yd 22
2
In data booklet
Variation of period of oscillation with mass for a spring
Solution to the SHM equation
ywdt
yd 22
2
wtAy cos
The expression for y which fits requires
advanced calculus so we will limit ourselves to proving given solutions.
wtAy sin if y=0 at t=0
if y=A at t=0
In data booklet
In data booklet
With A maximum amplitude of Oscillation (from zero position)
A Zeroposition
Proof 1
wtAy sin
wtAwdt
dycos
wtAwdt
ydsin2
2
2
ywdt
yd 22
2
See data booklet for derivatives.
You need to be able to derive this!
Proof 2
wtAy cos
wtAwdt
dysin
wtAwdt
ydcos2
2
2
ywdt
yd 22
2
See data booklet for derivatives.
You need to be able to derive this!
Simple pendulum
L
y
yL
ga
θL
y
From s=rθ (circular motion) y=Lθ
with θ in radians.
Hence:
mg
θ
mgcosθ
T
mgsinθ
The restoring force is mgsinθ.Since F=ma, ma=-mgsinθ (as F and θ in opposite directions)
a=-gsinθ a=-gθ (if θ small)
(as ) L
y
As and for SHMyL
g
dt
yd
2
2
ywdt
yd 22
2
henceL
gw 2
HoweverT
w2
henceg
LT 2
You don’t need to be able to derive this!
Determination of “g”
g
LT 2
Velocity at a given time
wtAy sin
wtAwdt
dyv cos
A
ywt sin
Consider the solution
Then
Rearranging these equations gives
Aw
vwt cosand
Then: wtwtA
y
Aw
v 2222
sincos
12
2
22
2
A
y
wA
v
2
2
22
2
1A
y
wA
v
22222 ywwAv
22 yAwv
)( 2222 yAwv
See additional relationships in data booklet
You need to be able to derive this!
In data booklet.
Graphing SHMy
t
A
A
Zero position
0
-A
Graphing SHMy
t
A
A
Zero position
0
Graphing SHM v
A
Zero position
y
t
A
0
v
t0
Graphing v
A
Zero position
y
t
A
0
v
t0
Maximum velocity is at y=0.
Graphing a
A
Zero position
a
t0
ywa 2y
t
A
0
Graphing a
A
Zero position
a
t0
Maximum acceleration is at y=A but is opposite in direction.
ywa 2y
t
A
0
Energy and SHM
2
2
1mvEk
222
2
1yAwmEk
For mass shown:
)(2
1 222 yAmwEk In data booklet
A Zeroposition
Assuming no energy loss, the total energy in the system is equal to the maximum kinetic energy.
This is when v is at a maximum; when y=0.
So the total energy is 22
2
1Amw
Therefore, at displacement y the potential energy is:
ktotalp EEE
22222
2
1
2
1yAmwAmwEp
22
2
1ymwEp In data
booklet
Damping
t
y
In reality, SHM oscillations come to rest because energy is transferred from the system and so the amplitude of the oscillationdrops.This is called DAMPING.
Critical damping
t
y
If the energy transfer is such that there is no displacement pastthe zero point (assuming it starts at A) then the system is said to beCRITICALLY DAMPED.
However, it is possible to overdamp the system (with very large resistance) which means that the time it takes to stop moving isactually larger than if there was no artificial damping and thesystem is allowed to oscillate to rest.