SHM

Question 1

SHM of a spring equation

Answer 1

mẍ = -kx
(force = spring force)

Question 2

general SHM equation and solution

Answer 2

equation:
ẍ = -ω²x

solution:
x(t) = A*cos(ωt + ϕ)
ϕ: phase angle

Question 3

equation for the total energy of an oscillating spring

Answer 3

E= ½kA²
E: total energy
k: spring constant
A: amplitude

Question 4

equation for damped harmonic motion of a spring

Answer 4

mẍ + bẋ + kx= 0
k: spring constant
b: friction factor

Question 5

general damped harmonic motion equation

Answer 5

mẍ + γẋ + ω₀²x= 0
γ: the damping factor (Gamma symbol)

γ = b/m
ω₀ = (k/m)^½

Question 6

general displacement equations for the 3 types of damped oscillation

Answer 6

light: (ω₀ > γ/2)
x(t) = A* e^(-γ/2 * t) * cos(ωt +ϕ)

heavy: (ω₀ < γ/2)
x(t)= e^(-γ/2 * t) * [ Ae^(αt) + Be^(-αt) ]
α² = γ²/4 - ω₀²

critical: (ω₀ = γ/2)
x(t) = e^(-γ/2 * t) * (A+Bt)

A and B are unknowns based on the initial conditions

Question 7

quality factor equation

Answer 7

Q = ω₀/γ

Question 8

energy equation for very light damping

Answer 8

E = E₀ * e^(-γt)

Question 9

general equation for forced damped motion

Answer 9

mẍ + bẋ + kx= F₀*cos(ωt)

F₀*cos(ωt) is the driving force

Question 10

displacement equation for a forced oscillation

Answer 10

x = A*cos(ωt - δ)
ω: the driving frequency

where:
tan(δ) = γω/ (ω₀²-ω²)
A = F₀ / m[ (ω₀²-ω²)² + ω²γ² ]^½

Question 11

how does resonance work for a damped and undamped forced oscillator

Answer 11

undamped:
as ω –> ω₀, A –> ∞

damped:
resonance at ω = ω₀
δ = π/2 at resonance
δ transitions between 0 and π as ω increases

Question 12

equation of oscillation for an RLC circuit

Answer 12

Lq(..) + Rq(.) + 1/C q = V₀ * e^(iωt)
L: inductance
R: resistance
C: capacitance
q: charge

this gives
I = I₀ * e^(iωt)
I: current