A Byte of PhysicsLogo

Rolling constraints

Rolling without slipping is a kinematic constraint between translation and rotation. It states that the material point touching the surface has no tangential velocity relative to that surface. The relation reduces the number of independent velocities, but it is valid only while contact and sufficient static friction are maintained.

Contact-point statement

For a rigid body, the velocity of a point displaced by \(\mathbf r_c\) from the center of mass is

\[\mathbf v_c=\mathbf V_{\mathrm{cm}}+\boldsymbol\omega\times\mathbf r_c\]

If the surface point has velocity \(\mathbf v_s\), no slip requires the tangential part of relative velocity to vanish:

\[\left(\mathbf I-\hat{\mathbf n}\hat{\mathbf n}^{\mathsf T}\right)(\mathbf v_c-\mathbf v_s)=\mathbf0\]

Here \(\hat{\mathbf n}\) is the contact normal. The normal component is handled by contact and nonpenetration conditions. This vector form works for a moving belt as well as stationary ground.

import numpy as np

radius = 0.3
centre_velocity = np.array([2.0, 0.0, 0.0])
omega = np.array([0.0, 0.0, -2.0 / radius])
contact_velocity = centre_velocity + np.cross(omega, [0.0, -radius, 0.0])
print("Slip velocity (m/s):", contact_velocity[0])

Circular wheel on a flat stationary surface

For a wheel of radius \(R\) rolling along one straight axis, the magnitude relation is

\[v_{\mathrm{cm}}=R|\omega|\]

The signed relation depends on conventions. If rightward translation and counterclockwise rotation viewed from one side are both positive, a wheel rolling right has negative angular velocity, so

\[v_{\mathrm{cm}}=-R\omega\]

Derive the sign from contact geometry rather than memorizing it. Reversing the viewing axis reverses the angular sign while leaving the physical motion unchanged.

The corresponding displacement relation on a straight surface is

\[x-x_0=-R(\theta-\theta_0)\]

for that same convention. One full turn advances a distance equal to circumference \(2\pi R\). The angle must be unwrapped; a display angle that resets once per turn cannot determine total distance.

Degrees of freedom

Without rolling, horizontal position \(x\) and angle \(\theta\) are independent coordinates. The no-slip displacement equation removes one independent degree of freedom on a straight fixed surface. One coordinate can be stored and the other derived.

initial_angle, initial_position = 0.0, 0.0
angle, angular_velocity = -4.0, -2.0
position = initial_position - radius * (angle - initial_angle)
velocity = -radius * angular_velocity
print(f"Position: {position:.3f} m, velocity: {velocity:.3f} m/s")

This exact-coordinate approach prevents constraint drift for simple geometry. In complex three-dimensional contact, independent generalized coordinates may be harder to find, so constraint forces or impulses are solved instead.

Acceleration relation

For constant radius rolling along a straight stationary surface, differentiating the signed speed constraint gives

\[a_{\mathrm{cm}}=-R\alpha\]

This relates center acceleration along the surface to angular acceleration. It does not say the contact material point has zero acceleration. At the instant of contact that point has zero velocity relative to the surface, but its centripetal acceleration is generally nonzero.

On a curved path or with changing radius, differentiating the full vector constraint introduces changing tangent, normal, and contact-offset terms. Reusing

\(a=R\alpha\)

outside straight constant-radius geometry can omit them.

Static friction enforces, but does not guarantee, rolling

Static friction may provide the torque needed for rolling. Its magnitude adjusts to the required value up to a limit:

\[|f_s|\le\mu_sN\]

The equality sign applies only at impending slip, not during every rolling motion. A freely rolling wheel at constant speed on ideal level ground can have zero friction. A wheel accelerating under a drive or rolling down an incline can require nonzero static friction.

Solve the equations under the no-slip assumption, then check the required friction. If it exceeds the limit, discard that solution and use a sliding-contact model. Imposing the constraint despite insufficient friction creates an unphysical result.

Rolling down an incline

For a symmetric body rolling down an incline of angle \(\beta\), force and torque equations combine with the constraint to give

\[a=\frac{g\sin\beta}{1+I_{\mathrm{cm}}/(MR^2)}\]

This assumes a rigid body, fixed radius, stationary incline, no slipping, and negligible rolling resistance. The required friction is found from the same coupled equations and must pass the static limit.

Different shapes have different dimensionless inertia ratios. A hoop, disk, and solid sphere therefore accelerate differently even with the same mass and radius. The mass cancels from the ideal acceleration when inertia scales as \(MR^2\).

Static friction and energy

For ideal rolling on a stationary rigid surface, the contact point has zero instantaneous velocity relative to the ground. Static friction can therefore exert force and torque without doing work at the contact point. It redistributes the body's acceleration between translation and rotation while mechanical energy remains conserved when no other dissipative effects act.

This statement is not universal. A moving belt can transfer energy through static friction, and deformable tires exhibit rolling resistance and hysteresis. A contact patch is not always an ideal point. The energy conclusion follows from the velocity of the force application point in the selected frame.

Instantaneous center of zero velocity

For planar pure rolling on stationary ground, the contact point is the instantaneous center of zero velocity. At that instant, other point velocities can be viewed as a pure rotation about contact. The top of a rolling wheel then moves at twice the center speed:

\[v_{\mathrm{top}}=v_{\mathrm{cm}}+R|\omega|=2v_{\mathrm{cm}}\]

The contact point is not permanently the same material point; a new point takes its place as the wheel rolls. The instantaneous center method describes velocities at one instant and should not be treated as a fixed hinge for accelerations.

Slip detection and transition

Define a signed or vector slip residual from contact relative velocity. A nonzero value means the no-slip velocity constraint is violated. When sliding occurs, kinetic friction opposes relative tangential motion at contact, not necessarily center-of-mass velocity.

A rapidly spinning ball placed gently on a surface can experience friction that speeds up its center while slowing its rotation. The correct direction comes from contact-point relative motion.

mass, slope, mu_static = 2.0, np.deg2rad(20), 0.3
normal = mass * 9.81 * np.cos(slope)
required_friction = mass * 9.81 * np.sin(slope) / 3  # Solid disk.
print("Rolling is supported:", required_friction <= mu_static * normal)

The transition can involve static and kinetic coefficients, impact, or changing normal force. State the friction model and avoid rapid numerical switching near zero slip by using a physically justified contact rule.

Constraint residuals in numerical work

When translation and rotation are advanced independently, monitor

\[C_v=v_{\mathrm{cm}}+R\omega\]

and, for compatible straight-path initial conditions,

\[C_x=(x-x_0)+R(\theta-\theta_0)\]

Reducing time step should reduce unintended drift for a consistent method. Projecting the state back onto zero residual can hide the cause and change momentum or energy. Prefer dependent coordinates for simple constraints or calculate a constraint impulse whose effects are included in the physical ledger.

Changing display angle wrapping must not change either residual. Use unwrapped angle for the displacement constraint.

Multiple and three-dimensional contacts

A sphere rolling on a plane has two tangential no-slip constraints. A wheel constrained to point along its plane may also have a no-sideways-slip condition. These velocity constraints may be nonholonomic: they cannot always be integrated into one global relation among coordinates independent of path.

Multiple contacts can be redundant or incompatible. A rigid-body contact solver must handle normals, tangent directions, inertia tensor, and friction bounds together. The single scalar wheel equation is a special case, not a general contact algorithm.

Reliability checks

Rolling constraints reduce a coupled motion only while their contact and friction conditions hold. Starting from relative velocity at contact keeps the familiar wheel relations connected to their geometry and prevents them from being applied outside their domain.

A worked Python notebook

A wheel of radius 0.3 metres rolls on a stationary flat surface. The question for this calculation is precise: how does distance rolled change when we change clockwise angle magnitude, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

The distance-angle relation requires no slip and a fixed wheel radius. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from 0 to 12.5664 rad. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for clockwise angle magnitude, expressed in rad. The output is distance rolled, expressed in m. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = 0.3 q\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 6.28319 rad, the expression gives 1.88496 m. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("0.3*q", locals={"q": q})
reference_input = 6.283185307179586
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "rad")
print("Predicted distance rolled:", reference_output, "m")

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives 0 m. The midpoint, at 6.28319 rad, gives 1.88496 m. The upper endpoint gives 3.76991 m. Read these together with the assumptions: they describe different choices of clockwise angle magnitude within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([0.0, 6.283185307179586, 12.566370614359172])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(f"{input_value:.6g} rad -> {output_value:.6g} m")
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows clockwise angle magnitude; the vertical axis shows distance rolled. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(0.0, 12.566370614359172, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Clockwise angle magnitude (rad)")
ax.set_ylabel("Distance rolled (m)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does distance rolled change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 6.28319 rad is 0.3 in those ratio units. Locally, the output increases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = 0.3\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 6.346017160251382
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "m")
print("Full expression:", full_prediction, "m")
print("Approximation difference:", full_prediction - linear_prediction, "m")

Here the changed input is 6.34602 rad. The tangent estimate is 1.90381 m, while the full expression gives 1.90381 m. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 12.566370614359172
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of clockwise angle magnitude inside the stated interval, predict whether distance rolled will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: The distance-angle relation requires no slip and a fixed wheel radius. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of m: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(12.566370614359172) - evaluate(0.0))
for points in [101, 201, 401]:
    grid = np.linspace(0.0, 12.566370614359172, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

Sources

Share to: