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

FGravity

FSpring1

FSpring2

Fnet = ma

• Fnet = Fg + Fs1 + Fs2 = ma

• Fnet = Fhorizontal + Fvertical

• Let us assume the mass does not move up and down Fvertical = 0

• So, Fnet = Fhorizontal = FS-horizontal(1+2)

• Thus, ma = m(d2x/dt2) = -kx

Fnet = kx

ma = m(d2x/dt2) = -kx

• Let k/m =

(d2x/dt2) + 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

Xo

t

x

X(t)=Acos(t + )

t

a(t) = -Acos(t + )

ao

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/= 2m/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)kA2

Proof of E=(1/2)kA2

• Kinetic Energy = (1/2)mv2

V = SIN(t + )

(1/2)MASIN2 (t + )

• Elastic potential energy U=(1/2)kx2

x = Acos(t + )

U (1/2)kA2cos2./(t + )

E = K + U =

(1/2)kA2[sin2 (t + ) + cos2 (t + )]

= (1/2)kA2

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

• http://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=236• http://www.kettering.edu/~drussell/Demos/SHO/mass.html• http://farside.ph.utexas.edu/teaching/301/lectures/node136.html• http://www.physics.uoguelph.ca/tutorials/shm/Q.shm.html