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

Philosophy Of Physics

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16,671 görüntüleme • 1 ay önce •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

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)

Praveen Kumar

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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

ア

75,802 görüntüleme • 1 yıl önce

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

a=(x,y,o=mag(k=x/4-12.5,e=y/9)/9)=>point((q=x+99+cos(9/k)+ok(cos(e9)/3+cos(y/9)/.7)sin(o4-t)).7sin(c=oe/30-t/8)+200,200+y/9cos(c4-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

ア

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a=(x,y,k=x/12-12.5,d=abs(cos(k/2)+sin(y/2)*cos(k/5)/2+1))=>point((q=x/4+99+(d+0)*k*(.5+cos(d*2-t+y/8)))*.8*cos(c=y/69-t/8+d/6)+200,q/2*cos(c+d/3+y/69)+200) t=0,draw=$=>{t||createCanvas(w=400,w);background(0).stroke(w,46);for(t+=PI/30,i=4e4;i--;)a(i%300,i/300)}#つぶやきProcessing

a=(x,y,k=x/12-12.5,d=abs(cos(k/2)+sin(y/2)cos(k/5)/2+1))=>point((q=x/4+99+(d+0)k*(.5+cos(d2-t+y/8))).8cos(c=y/69-t/8+d/6)+200,q/2cos(c+d/3+y/69)+200) t=0,draw=$=>{t||createCanvas(w=400,w);background(0).stroke(w,46);for(t+=PI/30,i=4e4;i--;)a(i%300,i/300)}#つぶやきProcessing

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a=(x,y,d=mag(k=(4+cos(x/9-t))*cos(x/30),e=y/7-13)+sin(y/99+t/2)-4)=>point((q=3*sin(k*2)+sin(y/29)*k*(9+2*sin(cos(e)*9-d*4+t)))+40*cos(c=d-t)+200,q*sin(c)+d*35) t=0,draw=$=>{t||createCanvas(w=400,w);background(9).stroke(w,96);for(t+=PI/120,i=1e4;i--;)a(i,i/235)}#つぶやきProcessing

a=(x,y,d=mag(k=(4+cos(x/9-t))cos(x/30),e=y/7-13)+sin(y/99+t/2)-4)=>point((q=3sin(k2)+sin(y/29)k*(9+2sin(cos(e)9-d4+t)))+40cos(c=d-t)+200,qsin(c)+d35) t=0,draw=$=>{t||createCanvas(w=400,w);background(9).stroke(w,96);for(t+=PI/120,i=1e4;i--;)a(i,i/235)}#つぶやきProcessing

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This Lissajous Machine is straight-up sorcery. Two cranks at 33:37 ratio, linked arms, and x=3cos(33t), y=3sin(37t) weave a perfect lattice from thin air—no pixels, just raw harmonic motion. Trig functions don’t just describe beauty. They are the beauty.

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23,198 görüntüleme • 3 ay önce

a=(x,y,k=x/8-12.5,d=abs(cos(k/2)+sin(y/4)))=> point((q=x/4+90+(d+.1)*k*cos(d-t*2+y/24))*.7*cos(c=y/99-t/8+d*d/32)+200,(q+y)*.7*sin(c)+(q-y)/2*cos(c*2)+200) t=0,draw=$=>{t||createCanvas(w=400,w);background(0).stroke(w,36);for(t+=PI/60,i=4e4;i--;)a(i%200,i/300)}#つぶやきProcessing

a=(x,y,k=x/8-12.5,d=abs(cos(k/2)+sin(y/4)))=> point((q=x/4+90+(d+.1)kcos(d-t2+y/24)).7cos(c=y/99-t/8+dd/32)+200,(q+y).7sin(c)+(q-y)/2cos(c2)+200) t=0,draw=$=>{t||createCanvas(w=400,w);background(0).stroke(w,36);for(t+=PI/60,i=4e4;i--;)a(i%200,i/300)}#つぶやきProcessing

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25,819 görüntüleme • 1 yıl önce

a=(x,y,k=x/12-12.5,d=2*abs(cos(k/2)+sin(y/4)+cos(k/8)))=>point((q=x/4+90+(d)*k*cos(d-t*2+y/24))*.7*cos(c=y/79-t/8+d/8)+q/8*sin(c*4)+200,(q)*.7*sin(c)+200) t=0,draw=$=>{t||createCanvas(w=400,w);background(0).stroke(w,36);for(t+=PI/60,i=4e4;i--;)a(i%300,i/300)}//#つぶやきProcessing

a=(x,y,k=x/12-12.5,d=2abs(cos(k/2)+sin(y/4)+cos(k/8)))=>point((q=x/4+90+(d)kcos(d-t2+y/24)).7cos(c=y/79-t/8+d/8)+q/8sin(c4)+200,(q).7sin(c)+200) t=0,draw=$=>{t||createCanvas(w=400,w);background(0).stroke(w,36);for(t+=PI/60,i=4e4;i--;)a(i%300,i/300)}//#つぶやきProcessing

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15,336 görüntüleme • 1 yıl önce

a=(x,y,o=mag(k=x/4-12.5,e=y/9+9)/9)=>point((q=x+39+y+1/k+o*k*(cos(y)/4+cos(e))*sin(o*4-t))*.7*sin(c=o+e/99-t/8+i%2*3)+200,200+y/2*cos(c*4-o)-q/2*cos(c)) t=0,draw=$=>{t||createCanvas(w=400,w);background(6).stroke(w,46);for(t+=PI/60,i=4e4;i--;)a(i/2%100,i/350)}//#つぶやきProcessing

a=(x,y,o=mag(k=x/4-12.5,e=y/9+9)/9)=>point((q=x+39+y+1/k+ok(cos(y)/4+cos(e))sin(o4-t)).7sin(c=o+e/99-t/8+i%23)+200,200+y/2cos(c4-o)-q/2cos(c)) t=0,draw=$=>{t||createCanvas(w=400,w);background(6).stroke(w,46);for(t+=PI/60,i=4e4;i--;)a(i/2%100,i/350)}//#つぶやきProcessing

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a=(x,y,d=mag(k=(4+cos(y))*cos(x/4),e=y/8-20))=>point((q=sin(k*3)+sin(y/19+9)*k*(6+sin(e*14-d)))*cos(d/8+t/4)+50*cos(c=d-t)+200,q*sin(c)+d*7*sin(c/4)+200) t=0,draw=$=>{t||createCanvas(w=400,w);background(9).stroke(w,116);for(t+=PI/120,i=1e4;i--;)a(i,i/235)}//#つぶやきProcessing

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a=(x,y,d=mag(k=(4+cos(y))cos(x/4),e=y/8-20))=>point((q=sin(k3)+sin(y/19+9)k(6+sin(e14-d)))cos(d/8+t/4)+50cos(c=d-t)+200,qsin(c)+d7sin(c/4)+200) t=0,draw=$=>{t||createCanvas(w=400,w);background(9).stroke(w,116);for(t+=PI/120,i=1e4;i--;)a(i,i/235)}//#つぶやきProcessing

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a=(x,y,k=x/8-12.5,d=cos(k/2)+sin(y/3)-.5)=>point((q=x/4+60+d*k*(1+cos(d*4-t*2+y/14)))*.7*cos(c=y*d/169-t/8+d/9)+200+60*sin(t*3/32+c/4),(q+59)*.7*sin(c)+200) t=0,draw=$=>{t||createCanvas(w=400,w);background(0).stroke(w,36);for(t+=PI/30,i=4e4;i--;)a(i%200,i/400)}#つぶやきProcessing

a=(x,y,k=x/8-12.5,d=cos(k/2)+sin(y/3)-.5)=>point((q=x/4+60+dk(1+cos(d4-t2+y/14))).7cos(c=yd/169-t/8+d/9)+200+60sin(t3/32+c/4),(q+59).7*sin(c)+200) t=0,draw=$=>{t||createCanvas(w=400,w);background(0).stroke(w,36);for(t+=PI/30,i=4e4;i--;)a(i%200,i/400)}#つぶやきProcessing

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Try Grok Imagine The video generation is already getting wild You can generate cinematic shots, motion, atmosphere....all from a simple prompt

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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

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+okcos(y/8-t)sin(o4-t))/3atan(2sin(c=oe/50-t/8))+200,(ytan(c)+q)/3*cos(c)+200)} //#つぶやきProcessing

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50,717 görüntüleme • 1 yıl önce

a=(x,y,d=mag(k=5*cos(x/19)*cos(y/30),e=y/8-12)**2/59+2)=>point((q=4*sin(atan2(k,e)*9)+9*sin(d-t)-k/d*(9+sin(d*9-t*16)*3))+50*cos(c=d*d/7-t)+200,q*sin(c)+d*45-9) t=0,draw=$=>{t||createCanvas(w=400,w);background(9).stroke(w,36);for(t+=PI/240,i=1e4;i--;)a(i,i/41)}#つぶやきProcessing

a=(x,y,d=mag(k=5cos(x/19)cos(y/30),e=y/8-12)**2/59+2)=>point((q=4sin(atan2(k,e)9)+9sin(d-t)-k/d(9+sin(d9-t16)3))+50cos(c=dd/7-t)+200,qsin(c)+d*45-9) t=0,draw=$=>{t||createCanvas(w=400,w);background(9).stroke(w,36);for(t+=PI/240,i=1e4;i--;)a(i,i/41)}#つぶやきProcessing

ア

39,881 görüntüleme • 1 yıl önce

a=(x,y,d=5*cos(mag(k=x/8-12.5,e=y/8-12.5)/3))=>stroke(99+99/abs(k*sin(k))*sin(t*4+e)**2,96).point((x+(d*k+k*3)*sin(d*2.5-t))/2+150,d*13+(d*3+12)*(3+cos(d*3-t))+d*e+150) t=0,draw=$=>{t||createCanvas(w=400,w);background(6);for(t+=PI/60,i=4e4;i--;)a(i%200,i/200)} #つぶやきProcessing

a=(x,y,d=5cos(mag(k=x/8-12.5,e=y/8-12.5)/3))=>stroke(99+99/abs(ksin(k))sin(t4+e)**2,96).point((x+(dk+k3)sin(d2.5-t))/2+150,d13+(d3+12)(3+cos(d3-t))+d*e+150) t=0,draw=$=>{t||createCanvas(w=400,w);background(6);for(t+=PI/60,i=4e4;i--;)a(i%200,i/200)} #つぶやきProcessing

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