| Simple Harmonic Motion: | x(t) = Acos(ωt + Φ),
v(t) = dx(t)/dt = -ωAsin(ωt + Φ), a(t) = d2x(t)/dt2 = -ω2Acos(ωt + Φ) = -ω2x. |
| Energy: | K = ½mv2, U = ½kx2, E = K + U = ½kA2. |
| A mass on a spring: | ω = (k/m)½, T = 2π(m/k)½, f = (1/(2π))(k/m)½. |
| A simple pendulum: | θ(t) = θmaxcos(ωt + Φ), ω2 = g/L (small oscillations). |
| Mechanical waves: | wave equation: d2y(x,t)/dx2 = (1/v2)d2y(x,t)/dt2. |
| Sinusoidal waves: | y = Asin(kx ± ωt + Φ), k = 2π/λ, ω = 2π/T = 2πf, v = λ/T = λf. |
| Waves in a string: | v = (F/μ)½, F = tension in the string, μ = mass per unit length. |
| Standing waves: | String and tube with two open ends: fn = nv/(2L) = nf1. Tube with one closed end: fn = nv/(4L) = nf1, n = odd. |
| Doppler effect: | f = f0(v-vo)/(v-vs), f = observed frequency, f0 = frequency of source, v = wave velocity, vo = velocity of observer, vs = velocity of source. vo and vs are not the speeds, but the components of the observer's and the source's velocity in the direction of the velocity of the sound reaching the observer. |
| Polarizers: | Itransmitted = I0cos2θ. |
| Geometrical Optics | |
| Law of refraction: | n1sinθ1 = n2sinθ2 |
| Mirror equation: | 1/xo + 1/xi = 1/f, M = -xi/xo, f = R/2 |
| Lens equation: | 1/xo + 1/xi = 1/f, M = -xi/xo |
| Wave Optics | |
| Double slit, diffraction grating (maxima): | dsinθ = mλ, m = 0, 1, 2, ..., λ ≈ zd/(mL) |
| Single slit, (minima): | wsinθ = mλ, λ ≈ zw/(mL) |
| Resolving power: | θmin = 1.22 λ/D |
| Constructive interference (thin oil film on water): | 2noilt cosθt = (m + ½)λ, m = 0,1,2,... |
Photons are electrically neutral and always move with the speed of light.
Photon energy: E = hf.
Photon momentum: p = hf/c = h/λ.
Photoelectric effect: Energy of ejected electron = hf - φ, (φ = work function
of the metal).
deBroglie relations: E = hf, p = h/λ.
But the relationship between wavelength and energy is different for photons and
massive particles.
For photons: E = pc, therefore λ = hc/E.
For massive non-relativistic particles: E = p2/(2m), therefore
λ = h/√(2mE).