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 ——————-①

ω = 2π/T —————————— ②

① & ② ————— T = 2π√(m/k)

Describe Resistance in SHM

The resistance in an oscillating system causes Damping

State the relationship between Resonance and Amplitude

When and Oscillating system is Forced to oscillate at it’s Natural ƒrequency: The Amplitude is reaching it’s Max