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

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Philosophy Of Physics

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16,671 просмотров • 1 месяц назад •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.
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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.

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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)
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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)

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

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a=(x,y,o=mag(k=x/4-12.5,e=y/9)/9)=>point((q=x+99+cos(9/k)+o*k*(cos(e*9)/3+cos(y/9)/.7)*sin(o*4-t))*.7*sin(c=o*e/30-t/8)+200,200+y/9*cos(c*4-t/2)-q/2*cos(c)) t=0,draw=$=>{t||createCanvas(w=400,w);background(6).stroke(w,46);for(t+=PI/60,i=3e4;i--;)a(i%100,i/150)}#つぶやきProcessing

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t=0,draw=$=>{t||createCanvas(w=400,w);background(6).stroke(255,66);for(t+=PI/60,y=99;y<300;y+=3)for(x=99;++x<300;)o=mag(k=x/8-25,e=y/8-25)/4,point((q=x+y/3+k/cos(y/8)+1/k+o*k*cos(y/8-t)*sin(o*4-t))/3*atan(2*sin(c=o*e/50-t/8))+200,(y*tan(c)+q)/3*cos(c)+200)} //#つぶやきProcessing
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