PHYS 101 Lecture Notes - Lecture 29: Angular Frequency, Propagation Constant, Simple Harmonic Motion

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PHYS 101: Physics for the Life Sciences Lecture 29:
Simple Harmonic Motion
x(t)= Acos(ωt+δ)
V(t)=A ω sin(ωt +δ)
A(t)=A ω2sin (ωt +δ)
Acos(ωt +δ)¿2
PE=1
2k x2=1
2k¿
A ωsin (ωt +δ)¿2
KE=1
2mV 2=1
2m¿
Example:
Consider the following system (no friction)
Take as the system the spring and the mass
There is no friction, therefore friction does no work on the system.
The weight and normal force are perpendicular to the direction of displacement
and therefore they also do not work on the system.
Since there are no external forces acting on the system the Mechanical Energy is
conserved.
Emechanical=KE+PE+Ethermal
(
Ethermal =0
)
Acos (ω t +δ)¿2
A ωsin (ω t +δ)¿2+1
2k¿
¿1
2m¿
(Recall:
)
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Acos((
)t+δ)¿2
A(
)sin((
)t+δ)¿2+1
2k¿
¿1
2m¿
¿1
2k A2[si n2(
t+δ)+co s2(
t+δ)]
(Recall:
si n2θ+co s2θ=1
)
Emechanical=1
2k A2
Graphic Representation of KE and PE:
Plot of KE and PE vs time (at
t=0
the spring is fully stretched)
Red Line = PE
Blue Line = KE
We can also plot KE and PE vs x
Red Line = PE
Blue Line = KE
Aside:
PE=¿
the potential energy in the spring
¿1
2k x2
KE=1
2mV 2
(we need to write this in terms of x)
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