Simulation

Angular Position and Velocity · SimulatorSlope of the θ-t Graph Equals ω

Rotational MotionAngular kinematics

A spinning disk with adjustable angular velocity; θ-vs-t and ω-vs-t graphs render live, showing that ω is the slope of the position graph.

Published: September 9, 2026

Objective

Verify the kinematic law θ(t) = θ₀ + ω·t for constant angular velocity by reading θ directly from the live graph and confirming its slope equals ω. The simulation uses an idealized rigid disk rotating at constant ω, with no angular acceleration or bearing friction.

Setup

Set the angular velocity slider to 3 rad/s (default), the initial angle to 0°, and the disk radius to 1.2 m. Note the readouts: θ = 0.00 rad, ω = 3.00 rad/s, rim speed = 3.60 m/s.
Click Start and watch both graphs fill. After 10 seconds, pause the simulation and read the θ readout; it should read near 30.00 rad.
Click Reset, then change ω to 6 rad/s. Start again and observe that the slope of the θ-t line is visibly steeper, while the ω-t horizontal line sits at the new height.
Reset and set ω to −3 rad/s. Start and confirm the θ-t line has a negative slope and the ω-t line sits below the zero axis, corresponding to counter-clockwise rotation.
With any non-zero ω running, change the radius slider (after Reset) and verify the rim-speed readout changes while the θ-t and ω-t graphs are identical to the same ω without radius change.
Run a full 30-second trial at ω = 3 rad/s, then click Reset (which keeps the first run as a ghost) and run at ω = 6 rad/s to compare the two slopes on the graph panel.

The spinning disk at rest, with the reference spoke at the 45-degree starting angle set by the initial-angle slider.

After a full 30-second run at 3 rad/s, the theta-t graph shows a straight line whose slope equals the constant omega shown by the flat amber trace below.

With omega set to negative 3 rad/s, the theta-t slope is negative and the omega-t line sits below the axis, showing counter-clockwise rotation.

Analytical Prediction

For constant angular velocity, angular position grows linearly with time. With ω = 3 rad/s and θ₀ = 0 rad:

θ(t)=θ₀ + ω·t

=0 + 3.0 × 10

=30.0 rad (at t = 10 s)

At t = 30 s the disk completes:

revolutions=θ / (2π)

=90 / 6.283

≈14.32 full turns

Rim speed is independent of time but scales with radius. At r = 1.2 m:

v=|ω| · r = 3.0 × 1.2 = 3.60 m/s

Changing radius to 2.0 m at the same ω:

v=3.0 × 2.0 = 6.00 m/s

The θ-t graph slope (rise over run) should equal 30 rad / 10 s = 3 rad/s, identical to the ω readout.

Results Analysis

After a 10-second run at ω = 3 rad/s, the θ readout should show approximately 30.00 rad (within ±0.05 rad of the linear prediction). The θ-t line on the graph is a straight diagonal with slope 3 rad/s; the ω-t line is flat at height 3.00 on the amber trace. Pausing at any time t and dividing the θ readout by that t should recover ω to within the integration substep precision. Switching to ω = −3 rad/s should yield θ ≈ −30.00 rad at t = 10 s and an inverted graph. The rim-speed readout changes proportionally when the radius slider is moved, while both graphs remain unchanged, confirming that r does not enter the angular kinematic equations.

Source of Error

The simulation models a perfectly rigid disk rotating at exactly the slider's constant ω, with no angular acceleration, no bearing friction, and no moment of inertia effects. In a real spinning disk, friction and drive torque interact to produce transient angular acceleration before reaching steady state; those dynamics are absent here. The disk is also modeled as a point-mass rim with no distributed mass, so moment of inertia is not relevant. Because the physics is purely algebraic (θ = θ₀ + ω·t), forward-Euler integration introduces no accumulation error, and the residual gap between the prediction and the readouts is therefore purely from rounding at display precision, not from physical omissions.

Further Exploration

Set ω = 0 and click Start. Both graphs are flat lines. What does a zero slope on the θ-t graph mean physically? How does the disk behave compared to a non-zero ω?
Try θ₀ = 90° and ω = 3 rad/s. How does the θ-t graph's y-intercept change compared to θ₀ = 0°? Does the slope change? This isolates the intercept parameter from the rate parameter.
Compare ω = 5 rad/s and ω = −5 rad/s runs (using the ghost overlay). The two slopes are mirror images about the t-axis — how does this relate to clockwise versus counter-clockwise rotation on the disk?
Slide the radius to 0.5 m, then to 2.0 m, at ω = 3 rad/s (in Fresh state). The rim speed changes from 1.50 m/s to 6.00 m/s. Does the θ-t graph change at all? Why or why not?
At ω = 10 rad/s for 30 s, the disk completes about 47.7 full turns. Can you verify this from the final θ readout (θ ≈ 300 rad) divided by 2π?