From Circles to Spirals: What Happens When Speed Changes?

In our last post, we explored how a particle moves along an elliptical path, using a parametric position vector: r(t) = \ That model assumed a constant angular speed, ω (omega). But real life isn’t always so smooth. What happens when ω is not constant? In this post, we’ll explore how changing speed affects the motion of a particle. We’ll go from elegant ellipses to beautiful spirals, and we’ll simulate them using **Octave**, a free programming tool for doing math and science with code. ---

Motion with Changing Angular Speed

In our original setup, we had: x(t) = a*cos(ωt) y(t) = b*sin(ωt) If ω is constant, the motion traces out an ellipse over and over again. But what if we let the angular speed change over time — for example: ω(t) = ω₀ + k\*t This means that as time increases, the angle grows **faster** (if k > 0) or **slower** (if k < 0). The position becomes: r(t) = \ where θ(t) = ∫₀ᵗ ω(s) ds = ω₀\*t + (1/2)*k*t² Now we’re dealing with **non-uniform angular motion**, and the particle no longer repeats its path. Instead, it spirals outward or inward, depending on how ω changes. ---

Visualizing the Spiral with Octave

Let’s write some Octave code to simulate the spiral. In this example, we’ll assume: * a = 5 * b = 3 * ω₀ = 2π (1 full loop per second) * k = 4π (acceleration in angular speed) 
% Parameters
a = 5;
b = 3;
omega0 = 2 * pi;     % Initial angular speed
k = 4 * pi;           % Angular acceleration
t = linspace(0, 2, 1000);  % Time from 0 to 2 seconds

% Compute theta(t) = omega0*t + 0.5*k*t.^2
theta = omega0 * t + 0.5 * k * t.^2;

% Parametric spiral path
x = a .* cos(theta);
y = b .* sin(theta);

% Plot
plot(x, y);
axis equal;
title('Spiral Motion with Increasing Angular Speed');
xlabel('x');
ylabel('y');
---

What Happens in the Code?

* The angle θ(t) is growing faster than in the circular case. * That causes the particle to move faster around the ellipse — but because the angle is increasing non-linearly, the curve does not close. * Instead, each loop is slightly twisted and offset — the particle **spirals**. Try changing k to a negative value. You’ll see the spiral wind inward! ---

What Does This Teach Us?

  • When angular speed is not constant, ellipses become spirals.
  • This type of motion occurs in real-world systems — such as satellites losing speed or gaining momentum.
  • Math allows us to model and predict these fascinating motions precisely.
---

Try This Yourself:

  • Set k = 0 to recover elliptical motion.
  • Try omega0 = 0 and k > 0 to see a pure spiral from rest.
  • Plot theta(t) directly to understand how angular position grows over time.

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