What is oscillatory motion? Oscillatory motion occurs when a force acting on a body is proportional to the displacement of the body from equilibrium. F  x The Force acts towards the equilibrium position causing a periodic back and forth motion.

What are some examples? Pendulum Spring-mass system Vibrations on a stringed instrument Molecules in a solid Electromagnetic waves AC current Many other examples…

What do these examples have in common? Time-period, T. This is the time it takes for one oscillation. Amplitude, A. This is the maximum displacement from equilibrium. Period and Amplitude are scalers.

Forces Consider a mass with two springs attached at opposite ends… We want to find an equation for the motion. How should we start? Free-body diagram!!

Free body diagram F Gravity F Spring1 F Spring2

F net = ma F net = F g + F s1 + F s2 = ma F net =  F horizontal +  F vertical Let us assume the mass does not move up and down   F vertical = 0 So, F net =  F horizontal = F S-horizontal(1+2) Thus, ma = m(d 2 x/dt 2 ) = -kx

F net = kx

ma = m(d 2 x/dt 2 ) = -kx Let k/m =   (d 2 x/dt 2 ) +   x = 0 This is the second order differential equation for a harmonic oscillator. It is your friend. It has a unique solution…

Simple Harmonic motion The displacement for a simple harmonic oscillator in one dimension is… x(t) = Acos(  t +    is the angular frequency. It is constant.   is the phase constant. It depends on the initial conditions. What is the velocity? What is the acceleration?

Velocity: differentiate x with respect to t. dx/dt = v(t) = -  Asin(  t +  ) Acceleration: differentiate v with respect to t. dv/dt = a(t) = -   Acos(  t +  )

A T  XoXo t x X(t)=Acos(  t +  )

t a(t) = -   Acos(  t +  ) aoao AA T a t

Using data The accelerometer will give us all the information we need to confirm our analysis We can measure all the parameters of this particular system and use them to predict the results of the accelerometer.

What can we measure without the accelerometer? The mass, m Hooke’s constant, k That’s all! T = 2  /  = 2  m/k) 1/2 (Recall   = k/m) Everything else depends on the initial conditions. What does this tell us? The time period, T, is independent of the initial conditions!

Energy The system operates at a particular frequency, v, regardless of the energy of the system. v = 1/T = 2  (k/m) 1/2 The energy of the system is proportional to the square of the amplitude. E = (1/2)kA 2

Proof of E=(1/2)kA 2 Kinetic Energy  = (1/2)mv 2 V =   SIN(  t +  )   (1/2)M   A  SIN 2 (  t +  ) Elastic potential energy U=(1/2)kx 2 x = Acos(  t +  ) U  (1/2)kA 2 cos 2./ (  t +  )

E = K + U = (1/2)kA 2 [sin 2 (  t +  ) + cos 2 (  t +  )] = (1/2)kA 2

Damping Simple harmonic motion is really a simplified case of oscillatory motion where there is no friction (remember our FBD) For small to medium data sets this will not affect our results noticeably.

for the rest of class... We are going to find k and m and compare to the results of the accelerometer

Some cool oscillatory motion websites