Simple Harmonic Motions

What are the conditions for SHM

acceleration ∝ displacement

acceleration in the opposite direction of the motion

a ∝ -s

Derive eqⁿs for: x , v , a using Sinusoidal graph

x_o = amplitude

x = x_osin(ωt)

dx/dt = v = ωx_ocos(ωt)

dx²/d²t = dv/dt = -ω²[x_osin(ωt)] = a = -ω²x

Define these terms in terms of SHM

Rearrange velocity eqⁿ without t

v = ωx_ocos(ωt)

v² = ω²(x_o² - x_o²sin²(ωt))

x = x_osin(ωt)

v² = ω²(x_o² - x²)

v² = ω²(x₀² - x²)

v = ±ω√(x_o² - x²)

Relate the terms: k (spring constant) , m (mass) , ω (angular ƒrequency)

F = kx , F = ma , a = ω²r

F = mω²r = kx → k = (mω²r)/x

k = mω²

Interpret Interception of E_K and E_P

E_T

E_P

E_K

Derive E_T eqⁿ

E = Fx/2 = x(mω²x)/2 = E_T = mω²x_o²/2

Derive the relationship between T (Period) and m (Mass) in SHM

F = kx , F_T = -kx → a = (-kx)/m = -ω²r , (-k/m)x = (-ω²)x

T = 2π√(m/k)