7E Equations

Density

ρ = m/v

Pressure

P = F/A

Pressure at a depth in a fluid

P = P₀ + ρgh

Pascal’s Law

P = F₁/A₁ = F₂/A₂

Force at a depth

F = ½ ρgwh²

Continuity equation of a fluid

A₁v₁ = A₂v₂

Volume flow rate

dV/dt = (Δx A)/Δt = Av

Incompressible fluid

m₁=m₂

P₁ + ρgy₁ + ½ ρv₁² = P₂ + ρgy₂ + ½ ρv₂²

Frequency

f = N(1/t)

Angular frequency or oscillation

ω = 2πf = sqrt(k/m) = sqrt(mgd/i)

Period of motion

T = 2π/ω

Acceleration and angular frequency

a = -ωx

Spring constant

k = mg/x

Energy

E = ½ mv² + ½ kx² = ½ kA²

Oscillations - position

x(t) = Acos(ωt + φ)

Oscillations - velocity

v(t) = -Aωsin(ωt+φ)

Oscillations - acceleration

a(t) = -Aω²cos(ωt+φ)

Torsion constant

T = 2πsqrt(i/κ) = iω²

Speed of propagation (velocity for wave/oscillation problems)

v = λ/T = λf = ω/k

Traveling sinusoidal wave

y(x,t) = Asin(k(x-vt)) = Asin(kx-ωt)

What direction is wave motion in if the sin wave is (kx + ωt)?

-x direction

What direction is wave motion in if the sin wave is (kx - ωt)?

+x direction

Wave number

k = 2π/λ

Period

T = 2π/ω

Wave functions that move in -x direction

y(x,t) = Asin(-kx - ωt)

y(x,t) = Asin(kx + ωt)

y(x,t) = Acos(kx + ωt)

Wave speed on a rope

v² = T/κ = β/ρ = S/ρ = Y/ρ

(bulk modulus, shear modulus, young’s modulus)

Different speeds for waves/oscillations

Wave propagates at v = ω/k = λf = λ/T = sqrt(T/μ)

Medium oscillates at v_y = ωAsin(kx - ωt)

Energy of a small section of sinusoidally vibrating string section

(harmonic oscillator)

dE = ½ (dm)v_y² + ½ (dm) ω²y²

Power for a sinusoidally vibrating string

P = ½ μvω²A²

Intensity and distance

I₁r₁² = I₂r₂²

Intensity and power

I = P/(4πr²)

Incident wave

y₁(x,t) = Acos(kx - ωt)

Reflected wave

y₂(x,t) = Bcos(kx + ωt)

Is a reflected wave on a fixed end inverted or the same?

The reflected wave for a fixed end is inverted

Is the reflected wave on a free end inverted or the same?

The reflected wave for a free end is the same