S'H'M'

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S.H.M.

• KINEMATICS PERIODICITY

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Simple Harmonic Motion

• Motion which is reoccurring
• Mass on spring
• Horizontal
• Vertical
• Simple Pendulum

3
SHM Kinematics Graphs

• See IP demo
• BUNGIE JUMP SHM DEMO

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SHM Kinematics Graphs
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SHM Kinematics
POSITION
0 p/2
p 3/2p
2p

VELOCITY
-
0 p/2
p 3/2p
2p

ACCEL
-
0 p/2
p 3/2p
2p
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SHM Kinematics
POSITION
0 p/2
p 3/2p
2p

VELOCITY
-
0 p/2
p 3/2p
2p

ACCEL
-
0 p/2
p 3/2p
2p
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S.H.M. Terminology

• Period (T) time for one complete oscillation
  (cycle)
• Sec/cycle
• Frequency (f) number of cycles in a given time.
• Cycles/sec (Hertz) (Hz)
• f 1/T
• Amplitude (A) Magnitude of the displacement
  relative to equilibrium or center position.

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SHM PERIODICITYMass - Spring System

• See IP demo
• MASS SPRING PERIOD

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SHM SPRING-MASS
-A
A
0
Net force is attempting to return mass to
Equilibrium Position
A max displacement in x direction
- A max displacement in -x direction
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Fundamental of SHM
Frestore a -x
Frestore -kx
k spring constant
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SHM SPRING-MASS
-A
A
SOLVE FOR MAXIMUM VELOCITY
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SHM SPRING-MASS
0
-A
A
SOLVE FOR MAXIMUM VELOCITY
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SHM SPRING-MASS
0
-A
A
SOLVE FOR VELOCITY AT ANY POINT
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Derive Period (T) Equation
T ?
T vs Mass?
T vs Amplitude?
T vs k (spring constant)?
T vs gravity?
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The shadow cast by mass in circular motion
matches motion of mass oscillating on spring
R A
Mass in uniform circular motion at constant speed
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PE spring 1/2 kDx2
PE spring 1/2 kA2
at x 0, KE mass PE spring
1/2 mvmax2 1/2 kA2
vmax2 (k/m)A2
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Relate mass-spring to circular motion
Speed distance / time
V C/T
T C / V
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T 2pA
T 2p
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Mass - Spring System
T ind A
T ind g
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Mass Spring SHM
A
- A
Net force is attempting to return mass to
Equilibrium Position
Restoring Force
A max displacement in x direction
- A max displacement in -x direction
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Simple Pendulum

• See IP demo
• CC

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Fundamental of SHM
Frestore a -x
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SIMPLE PENDULUM
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Derive Period (T) of Pendulum
T ?
T vs Mass?
T vs Amplitude?
T vs length?
T vs gravity?
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Simple Pendulum
Fc 0
v 0
Wrad T
q
L
Net F Wtan -mgsin q
tan
T
Net F a sin q, not q
A
(Not true SHM)
Wrad
If q small, sin q q
q
Wtan
W
sin q A/L
Net F mgA/L
rad
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Relate pendulum to circular motion
Speed distance / time
V C / T
T C / V
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Net F is Center Seeking F
Net F Fc
Net F mgA/L
Fc mv2/R
mv2/R mgA/L
(RA)
v2 (g/L)A2
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T 2pA
T 2p
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Simple Pendulum
T ind mass
T ind q
(for small angles)