Circular Motion & Angular Acceleration | Interactive Physics

Interactive experiment

🟣 Trajectory🔵 Velocity vector🟠 Inward radial direction⚫ Centre

Initial angular speed ω₀ (rad/s)Angular acceleration α (rad/s²)Initial radius r₀ (m)Trajectory typeRadial decrease k (m/s)

Elapsed time0.00 s

Angular position θ0.0°

Angular speed ω1.20 rad/s

Radius r5.00 m

Linear speed |v|6.00 m/s

Angular acceleration α0.20 rad/s²

Select a trajectory and press Start motion.

Observation: A positive α makes counterclockwise rotation speed up when ω is positive. A negative α slows it down and may reverse the direction. In spiral mode, the particle also moves radially inward, so its velocity is not purely tangential.

Python Mathematical Model

This independent Python example calculates position, angular speed and linear velocity using the same equations as the browser simulation. Change the input values and compare.

import math

omega0 = 1.2    # initial angular speed (rad/s)
alpha = 0.2     # angular acceleration (rad/s^2)
r0 = 5.0        # initial radius (m)
k = 0.35        # radial decrease (m/s)
spiral = True   # False = fixed-radius circle

def state(t):
    theta = omega0*t + 0.5*alpha*t*t
    omega = omega0 + alpha*t
    r = max(0.0, r0-k*t) if spiral else r0
    rdot = -k if spiral and r > 0 else 0.0
    x = r*math.cos(theta)
    y = r*math.sin(theta)
    vx = rdot*math.cos(theta)-r*omega*math.sin(theta)
    vy = rdot*math.sin(theta)+r*omega*math.cos(theta)
    speed = math.hypot(vx, vy)
    return x, y, theta, omega, r, speed

for t in [0, 1, 2, 3, 4, 5]:
    x, y, theta, omega, r, v = state(t)
    print(f't={t}s: x={x:.2f}, y={y:.2f}, '
          f'omega={omega:.2f}, r={r:.2f}, v={v:.2f}')

# This is a prescribed kinematic model,
# not a gravitational orbital simulation.

Understanding Angular Motion

Angular speed describes how rapidly the angle changes, measured in radians per second. Angular acceleration describes how rapidly angular velocity changes, measured in radians per second squared.

For a fixed radius, the tangential speed is |v| = r|ω|, tangential acceleration is aₜ = rα, and inward centripetal acceleration has magnitude aᵣ = rω².

For the spiral, radial velocity is dr/dt = −k. Its radial acceleration is −rω² and its transverse acceleration is rα + 2(dr/dt)ω, since the radius is changing.

Satellite analogy: The shrinking-radius path illustrates inward spiralling qualitatively. Real orbital decay depends on gravity, drag, energy and angular momentum; a satellite does not normally follow a constant radial decrease or constant angular acceleration.

Challenge: Set α = 0, then double ω₀. What happens to the time for one revolution? Next switch to spiral mode and observe how the velocity vector tilts inward.