OSCILLATIONS

- Definitions

- Any motion that repeats itself in equal periods of time - Periodic Motion
- If the motion can be described in terms of sines and cosines - Harmonic Motion
- Motion that is back and forth over the same path - Oscillatory
- Period (T) - time for a round trip
- Frequency: f = 1/T
- Equilibrium position - no net force
- Displacement is measured with respect to the equilibrium position

- One-dimensional motion

- General ideas

- At equilibrium, x = 0, and there is no unbalanced force, F = 0
- If the forces are conservative, the total mechanical energy is constant

E = KE + PE = constant

- For a mass attached to a spring

F = -kx

k = spring constant (Hooke's Law)

- This simple harmonic oscillator (SHO) undergoes simple harmonic motion (SHM)
- We can associate a potential energy (PE) with a SHO
- Since,
- Solution,
- Substituting this into the original differential equation we find w
^{2}= k/m - Motion repeats itself after a time of 2p /w and
- Energy considerations

F = -kx = -(dU/dx)

dU = kxdx

PE =U

U = (1/2)kx^{2}

(after an integration)

F_{NET} = ma = m(dx^{2}/dt^{2})= -kx

m(dx^{2}/dt^{2}) + kx = 0

x = Acos(w t + d )

where A, w , and d are constants

T = 2p /w = 1/f

f = (1/2p
)(k/m)^{(1/2)}

x = Acos(w t + d )

U = (1/2)kx^{2} = (1/2)k[Acos(w
t + d
)]^{2}

KE = (1/2)mv^{2} = (1/2)m(dx/dt)^{2}

KE = (1/2)mw
^{2} [Asin(w
t + d
)]^{2}

E = KE + PE

E = (1/2)kA^{2}

(How did we get this?)

Also,

v = dx/dt = ±
[(k/m)(A^{2}-x^{2})]^{(1/2)}