# 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:

ω=

√

❑

)

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)