Rolling constraints
Rolling without slipping ties linear and angular motion together. The contact point has zero speed relative to the ground, so one degree of freedom determines the other.
Think like a programmer
Model checklist
- Inputs
- Wheel radius, position, angular position, and one chosen speed.
- State
- Position and orientation constrained by contact.
- Rule
- Derive the paired speed from the no-slip relation.
- Output
- A wheel that covers the correct distance per turn.
- Check
- One full turn moves a wheel by its circumference.
Torque updates angular motion
Change torque and moment of inertia. The same torque produces less angular acceleration for a harder-to-rotate body.
Torque 6 N·m; moment of inertia 2 kg·m²; angular acceleration 3.00 rad/s²; rotational energy at 3 rad/s 9.00 J.
Keep one independent coordinate
For a wheel of fixed radius (R) on a stationary flat surface, choose unwrapped rotation (\theta) or horizontal position (x) as the independent state. The holonomic relation is:
\[x-x_0=R(\theta-\theta_0), \qquad v=R\omega, \qquad a=R\alpha.\]Derive the paired quantity after each update from the same old-to-new state; do not integrate both freely and snap them together later. Snapping hides drift and can introduce an unexplained energy or momentum change. Keep display-angle wrapping separate from the unwrapped solver angle.
Define the contact frame and sign convention. The no-slip statement is that the instantaneous velocity of the contact point relative to the surface is zero, not that friction is absent. Static friction may be the constraint force that makes the relation hold and can redistribute translational and rotational energy without dissipating it in the ideal case.
Detect a violated constraint
Log the residual (c=(x-x_0)-R(\theta-\theta_0)), the contact-point relative speed, radius, and any tangential force. Test one turn advancing by (2\pi R), reversed rotation reversing translation, a zero-radius rejection, and constant angular velocity at fixed radius. Under a force model, compare translational-plus-rotational energy and the work of external forces at smaller steps.
If required static friction exceeds the available (\mu_sN), switch explicitly to a sliding model: relative contact speed is no longer zero, kinetic friction has direction and dissipative work, and (v=R\omega) must not be imposed. Ramps, deformable tyres, finite contact patches, and rolling resistance need richer geometry and force laws.
Try this experiment
Prediction: Doubling the radius at fixed angular speed doubles the translation speed.
Use the equation to predict the distance a wheel travels in ten radians, then compare it with its circumference.Where this model breaks
Summary
Encode no-slip rolling as an explicit contact constraint. Advance one unwrapped coordinate, derive the other, measure the constraint residual, and switch models rather than silently enforcing the relation when sliding begins.
Glossary
- No-slip condition: contact point is instantaneously at rest relative to the surface.
- Angular speed: rate of change of orientation.
- Constraint: a rule limiting valid states.
Self-check
- Which value can you derive from angular speed?
- What distance follows from one turn?
- What physical event breaks the relation?
Sources
- OpenStax, University Physics Volume 1, rolling motion.
Model contract
Treat the lesson as a small function before treating it as a fact to memorize. Give every value a unit, keep only the state needed for the next step, and make the output easy to inspect.
\[\text{observable output} = f(\text{inputs},\,\text{state})\]- Inputs
- Quantities you set or measure, with units and useful bounds.
- State
- Values the program must retain to reproduce the next result.
- Rule
- The relationship or update that turns inputs and state into a result.
- Check
- A known limit, unit check, invariant, or measured result that can expose a bad model.
Implement the idea as a model
For Rolling Constraints, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Rolling, Angular Momentum, and Gyroscopes, the useful program is not the drawing: it is the smallest explicit model that makes a prediction you can test.
Guided experiment
Prediction: changing one declared input while holding the others fixed should change only the outputs that the model connects to that input. Choose one input, predict the direction of change, then check a limiting case such as zero, a symmetric arrangement, or a familiar low-speed or small-change approximation.
Where this model breaks
This lesson is a teaching model, not a complete simulator. Before using it outside the stated question, check which interactions, scales, uncertainties, boundary conditions, and measurement limits it leaves out.
Summary
Treat Rolling Constraints as a contract: named inputs and units enter a rule, the rule produces an observable result, and a known limit or invariant checks whether the implementation deserves trust.
Glossary
- Input: a measured value or chosen parameter supplied to a model.
- State: the smallest set of values needed to continue or reproduce a model.
- Validation: comparing an output with a known result, limit, invariant, or measurement.
Self-check
- Which values are inputs, and which values must remain state?
- What observable result would tell you the model is behaving as expected?
- Which assumption would you test first before applying the model to a real system?
Model review: turn Rolling Constraints into a test
Model no-slip rolling as a state constraint that couples wheel translation and rotation.
- Name the inputs and units that the rolling, angular momentum, and gyroscopes model needs.
- Separate the state you must keep from values you can calculate when needed.
- Write one rule that maps the current state and inputs to an observable result.
- Choose a limiting case, unit check, invariant, or known result before trusting an output.
- State one assumption you would change before using this simplified model for a real decision.