Simple Harmonic Motion: periodic motion that is a sinusoidal function of time

F =maF = −kxma = −kx

a = −kxm

Linear Simple Harmonic Oscillator Force in SHM

a  is  greatest  when  x  is  greatest,  but  in  opposite  direc4on.    When  x  is  0,  a  is  0.  

Energy  in  SHM    (horizontal  mass-­‐spring  system)  

Etotal =12mv2 + 1

2kx2

Etotal = 0+12kA2

Etotal =12kA2

At  maximum  displacement  from  equilibrium  (x  =  Amplitude):  

12mv2 + 1

2kx2 = 1

2kA2

12mv2 = 1

2kA2 − 1

2kx2

v = ± km(A2 − x2 )

Combine  to  get  equa4on  for  velocity  as  a  func4on  of  posi4on:  

Etotal =12kA2

Etotal =12mvmax

2

12kA2 = 1

2mvmax

2

Avmax

=mk

vmax =2πAT

T = 2πAvmax

T = 2π Avmax

= 2π mk

τ = −L(mgsinθ )−L(mgsinθ ) = Iα

α = −mgLIsinθ

SIMPLE PENDULUM

Restoring Torque:

To be true SHM, α should be proportional to θ rather than sin θ. However, θ and sin θ are approximately equal as long as θ is small. (This is known as the small angle approximation.)

SIMPLE PENDULUM:

Period

T = 2π ℓg