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Simple Harmonic Motion visualized from the formula x(t) = A cos(ωt + φ)

Philosophy Of Physics

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16,701 次观看 • 2 个月前 •via X (Twitter)

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This video looks like a rotating wheel. But watch just one single sphere. It never rotates. Every ball only moves in a perfectly straight line back and forth. This is simple harmonic motion where linear position follows: x = R cos(ωt) By perfectly offsetting the timing of each straight path, the collective motion creates the illusion of rotation.

This video looks like a rotating wheel. But watch just one single sphere. It never rotates. Every ball only moves in a perfectly straight line back and forth. This is simple harmonic motion where linear position follows: x = R cos(ωt) By perfectly offsetting the timing of each straight path, the collective motion creates the illusion of rotation.

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This fascinating experiment reveals how simple harmonic motion can be generated from uniform circular motion

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1. Spatial Constraints & Magnitude: K(u) = A₁ * cos(u / λ₁) E(v) = (v / λ₂) - δ_offset Distance Driver: D(u,v) = √( K(u)² + E(v)² ) 2. Pulsation Kinematics: Q(u,v,t) = αsin(2K) + (β/K) + sin(v/λ₃) * K * [γ + εsin(ω₁E - ω₂D + ω₃t)] ↳ The (β/K) term creates the asymptotic stretch of the tentacles, while the nested sine waves act as a damped harmonic oscillator for the contraction of the bell. pause - push - pause - push (action) 3. Polar-to-Cartesian Mapping Rotational Phase: θ(u,v,t) = D(u,v) - t X(u,v,t) = Q * cos(θ) * S_x + C_x Y(u,v,t) = Q * sin(θ) + S_y * D - C_y 4. Particle Sizing Filter: Radius = K(u)² > τ ? 2 : 1 Alpha-Blended Buffer: C_final = (1 - ρ) * C_new + ρ * C_buffer (u,v = particle indices, t = continuous time, ρ = retention rate for trails)

1. Spatial Constraints & Magnitude: K(u) = A₁ * cos(u / λ₁) E(v) = (v / λ₂) - δ_offset Distance Driver: D(u,v) = √( K(u)² + E(v)² ) 2. Pulsation Kinematics: Q(u,v,t) = αsin(2K) + (β/K) + sin(v/λ₃) * K * [γ + εsin(ω₁E - ω₂D + ω₃t)] ↳ The (β/K) term creates the asymptotic stretch of the tentacles, while the nested sine waves act as a damped harmonic oscillator for the contraction of the bell. pause - push - pause - push (action) 3. Polar-to-Cartesian Mapping Rotational Phase: θ(u,v,t) = D(u,v) - t X(u,v,t) = Q * cos(θ) * S_x + C_x Y(u,v,t) = Q * sin(θ) + S_y * D - C_y 4. Particle Sizing Filter: Radius = K(u)² > τ ? 2 : 1 Alpha-Blended Buffer: C_final = (1 - ρ) * C_new + ρ * C_buffer (u,v = particle indices, t = continuous time, ρ = retention rate for trails)

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a=(x,y,d=5*cos(o=mag(k=x/8-25,e=y/8-25)/3.5))=>[(k*atan(3*cos(d*9))+x/2)*cos(c=(d+o)/4-t/8)+200,(k*d*cos(o-d+t)+y/2)*sin(c)+200] t=0,draw=$=>{t||createCanvas(w=400,w);background(6).stroke(255,96);for(t+=PI/60,y=99;y<300;y+=2)for(x=99;++x<300;)point(...a(x,y))} #つぶやきProcessing

a=(x,y,d=5cos(o=mag(k=x/8-25,e=y/8-25)/3.5))=>[(katan(3cos(d9))+x/2)cos(c=(d+o)/4-t/8)+200,(kdcos(o-d+t)+y/2)sin(c)+200] t=0,draw=$=>{t||createCanvas(w=400,w);background(6).stroke(255,96);for(t+=PI/60,y=99;y<300;y+=2)for(x=99;++x<300;)point(...a(x,y))} #つぶやきProcessing

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