string, frequency, antinodes:

Ends of a stretched string are called nodes, N. Point of highest vibrational amplitude is antinode, A. Number of N-1 = harmonic full sine wave (2N) = wavelength

Example:

Adjacent antinodes are separated by a distance of 20 cm and waves travel at a speed of 1200 cm/s along the string. 
What frequency is the string vibrating at?
  wavelength = 40 cm
  freq = v/wavelength, v = velocity of wave
  1200/40 = 30

Doppler effect ⇒ F = Fo (v +/- vo) / (v +/- vs)

v = velocity (343 m/s for speed of sound)
vo = velocity of observer (negative if moving away from source)
vs = velocity of source (negative if approaching observer)

Hooke's law for springs: F=-kx, k=spring constant, x = displacement

Spring energy:

• potential energy: U = 1/2 kx², or
• P = mgh (at mass at a given height)
• kinetic energy: K = 1/2 mv²

conservation of energy: potential energy + kinetic energy = constant, for a system

Fulcrum: t = r * f (torque = radius * force) just add the torques for multiple objects on one side of a fulcrum

F = ma ⇒ ma = -kx

angular frequency ω = √(k/m)

period of oscillation T = 2π √(m/k) (horizontal or vertical springs)

In a vertical spring, the weight Mg of the body produces an initial elongation to equilibrium, such that Mg − kyₒ = 0.

If y is the displacement from this equilibrium position the total restoring force will be Mg − k(yₒ + y) = −ky

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