module 11 oscillatory and simple harmonic motion

spring acceleration

a_x = -k/m • x = d²x/dt²

x(t) =

Acos(ωt + φ); A = amplitude, ω = angular freq, phi = phase constant

solve for unknown phase constant by

plugging in t = 0 and using both x(0) and v(0) equations

ω =

√k/m = 2πf = 2π/T - frequency in rads/s, rather than full rotations

φ

phase constant → adjusts starting time

f

1/2π • √k/m

T

2π • √m/k

v(t) =

-Aωsin(ωt + φ)

V_max =

Aω

a(t) =

-Aω²cos(ωt + φ)

total energy =

½ kA²

total energy as a function of position

½ kx² + ½ mv²

velocity as a function of position

= √(k/m)(A² - X²) = Aω√1 - (x²/A²) = V_max√1 - (x/A)²

T of pendulum

2π • √L/g

θ of physical pendulum =

θ_maxcos(ωt + φ)

period of something swinging?

T = 2π√I_p/mgd

drag from damping

-bv_x

x(t) when damped

Ae^-(b/2m)tcos(ωt + φ)

ω when damped 

√(k/m) - (b/2m)²

ω when damped

√(k/m) - (b/2m)²

underdamped when

b/2m < √k/m

critically damped when

b/2m = √k/m

overdamped when

b/2m > √k/m

see damping graph

in notes