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Constraint forces

A constraint force is a physical interaction that prevents a body from entering a forbidden configuration. A table's normal force prevents a book from passing through the table. Tension keeps an ideal taut rope from changing length. A rail pushes a cart toward its allowed track. The force is not chosen arbitrarily: its magnitude follows from Newton's laws together with the geometric constraint.

Geometry and dynamics are separate equations

For a smooth equality constraint

\[C(\mathbf r,t)=0\]

allowed velocities satisfy

\[\nabla C\cdot\mathbf v+\frac{\partial C}{\partial t}=0\]

and allowed accelerations follow by differentiating again. Newton's second law supplies the force relationship:

\[m\mathbf a=\mathbf F_{\mathrm{applied}}+\mathbf F_{\mathrm{constraint}}\]

The geometric equation does not by itself say which materials produce the reaction, whether contact can pull, or whether energy is dissipated. Those facts belong to the contact, joint, rope, or rail model.

Normal contact force

A normal force is perpendicular to the local contact surface. For a book resting on a horizontal table, choose upward positive. If the book has no vertical acceleration and weight is the only other vertical force,

\[N-mg=0\quad\Longrightarrow\quad N=mg\]

This equality is a result of this particular force inventory and acceleration, not the definition of normal force. In an elevator with upward acceleration \(a_y\),

\[N-mg=ma_y\quad\Longrightarrow\quad N=m(g+a_y)\]

The scale reading is normal force, sometimes called apparent weight. During free fall, \(a_y=-g\) and \(N=0\) because contact is lost. Gravity has not disappeared.

Ordinary passive contact is unilateral: the surface can push but cannot pull. Its normal magnitude therefore satisfies

\(N\ge0\). A calculation demanding negative normal force means the assumed contact cannot be maintained without adhesion or another interaction.

Curved paths require normal acceleration

For speed \(v\) along a path with local radius of curvature \(R\), the normal acceleration magnitude is

\[a_n=\frac{v^2}{R}\]

At the top of a convex circular hill, choose inward toward the circle center as positive. For a vehicle with weight and road normal force,

\[mg-N=m\frac{v^2}{R}\]

so

\[N=m\left(g-\frac{v^2}{R}\right)\]

Contact requires this value to be nonnegative. The ideal contact threshold is \(v=\sqrt{gR}\). Above it, the road would need to pull downward, so an unsecured vehicle leaves the surface and follows free flight. Clamping it to the path would violate the contact model.

mass, gravity, speed, radius = 2.0, 9.81, 4.0, 3.0
required_normal = mass * (gravity - speed**2 / radius)
normal_force = max(required_normal, 0.0)
contact_mode = "touching" if required_normal >= 0 else "separated"
print(f"{contact_mode}: normal force {normal_force:.3f} N")

This mode decision assumes a smooth hill, no downforce, and the stated inward axis.

Inclined contact

For a block on a fixed incline at angle \(\theta\), the perpendicular component of weight is \(mg\cos\theta\). With no other normal force and no normal acceleration,

\[N=mg\cos\theta\]

An applied force with a component into the surface increases the required normal force; a component away from it decreases the force and can cause separation. Friction limits depend on the resulting normal force, so calculating friction before the normal balance can create a circular or incorrect assumption.

Static friction as a constraint force

Static friction enforces no relative sliding when the required tangential force lies within a bound:

\[|f_s|\le\mu_sN\]

The actual static friction is whatever value satisfies the tangent equation, up to this limit. It is not automatically \(\mu_sN\).

applied_horizontal_force = 5.0
normal_force = 2.0 * 9.81
mu_static, mu_kinetic = 0.4, 0.3
required_friction = -applied_horizontal_force
static_limit = mu_static * normal_force
if abs(required_friction) <= static_limit:
    mode, friction = "sticking", required_friction
else:
    mode, friction = "sliding right", -mu_kinetic * normal_force
print(mode, f"friction = {friction:.3f} N")

At exactly zero sliding velocity, direction must come from the tendency to slide and the coupled equations, not from dividing the zero vector by its magnitude.

Tension and inextensible ropes

An ideal taut inextensible rope constrains its length. In the simplest two-end geometry,

\[|\mathbf r_2-\mathbf r_1|=L\]

Tension pulls each attached body along the rope toward the other attachment or pulley. An ordinary rope does not push, so \(T\ge0\). A negative solved tension means the taut-rope assumption is inconsistent and the rope should become slack.

Uniform tension throughout a rope is an approximation requiring a massless rope and ideal pulley conditions. A pulley with rotational inertia obeys

\[(T_2-T_1)R=I\alpha\]

so the tensions generally differ when angular acceleration and inertia are nonzero. Replacing them with one shared value would erase the torque that accelerates the pulley.

Lagrange multiplier form

For an ideal smooth holonomic constraint, a reaction can be represented along the constraint gradient:

\[m\ddot{\mathbf r}=\mathbf F_{\mathrm{applied}}+\lambda\nabla C\]

The multiplier \(\lambda\) is solved so the acceleration remains compatible with the constraint. Its numerical value depends on how \(C\) is scaled; the physical force product \(\lambda\nabla C\) does not.

For several constraints, collect their gradients in a Jacobian and solve the coupled system for all multipliers. Redundant or dependent constraints can make this solve singular. A solver should report residual and rank problems rather than silently choosing arbitrary reactions.

Contact impulse is not continuous force

When bodies collide, velocity may change over a time too short for a rigid-body model to resolve. An impulse

\[\mathbf J=\int\mathbf F\,dt=\Delta\mathbf p\]

is then more appropriate than reporting the position correction as a force. Dividing an impulse by the simulation step produces a step-dependent average force and should not be called the peak physical contact force.

Sustained contact and impact are different modes. Sustained contact solves a reaction that prevents normal acceleration into the surface. Impact applies an impulse that changes normal velocity. A complete contact algorithm decides activation, impact, sticking or sliding, and separation.

Why clamping is incomplete

Clamping a penetrated position to a boundary enforces a visible geometry but leaves questions unanswered:

A small projection can control accumulated numerical residual after a physically defined solve. A large silent projection is not evidence that the constraint force was calculated.

Constraint-force calculation pattern

def support_reaction(mass, upward_applied_force, gravity=9.81):
    required = mass * gravity - upward_applied_force
    reaction = max(0.0, required)
    acceleration = (upward_applied_force + reaction - mass * gravity) / mass
    return reaction, acceleration, required >= 0


for pull in [0.0, 10.0, 30.0]:
    reaction, acceleration, touching = support_reaction(2.0, pull)
    print(pull, reaction, acceleration, touching)

Every residual needs units and a stated tolerance. Position, velocity, and force tolerances cannot all be the same raw number.

Work and ideal constraints

For a fixed smooth surface, an ideal normal force is perpendicular to allowed tangent velocity, so its instantaneous power is

\[P_C=\mathbf F_C\cdot\mathbf v=0\]

This does not mean every constraint force does no work. Friction can dissipate energy, a moving boundary can transfer energy, and an inelastic impact can reduce translational kinetic energy. The zero-work statement needs the fixed ideal constraint assumptions.

Validation checks

Constraint forces are not arbitrary corrections. They are forces or impulses supplied by physical interactions, solved together with Newton's laws and the geometry that the system is required to obey.

A worked Python notebook

A 2 kilogram body travels over a convex crest of radius 3 metres. The question for this calculation is precise: how does required normal reaction change when we change speed at a hill crest, 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.

A negative required reaction means loss of contact, so the contact model cannot continue beyond that threshold. 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 5 m/s. 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 speed at a hill crest, expressed in m/s. The output is required normal reaction, expressed in N. 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.

\[f(q) = 19.62 - \frac{2 q^{2}}{3}\]

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 4 m/s, the expression gives 8.95333 N. 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("2*(9.81-q**2/3)", locals={"q": q})
reference_input = 4.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "m/s")
print("Predicted required normal reaction:", reference_output, "N")

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 19.62 N. The midpoint, at 2.5 m/s, gives 15.4533 N. The upper endpoint gives 2.95333 N. Read these together with the assumptions: they describe different choices of speed at a hill crest 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, 2.5, 5.0])
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} m/s -> {output_value:.6g} N")
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 speed at a hill crest; the vertical axis shows required normal reaction. 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, 5.0, 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("Speed at a hill crest (m/s)")
ax.set_ylabel("Required normal reaction (N)")
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 required normal reaction change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 4 m/s is -5.33333 in those ratio units. Locally, the output decreases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = - \frac{4 q}{3}\]

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 = 4.025
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, "N")
print("Full expression:", full_prediction, "N")
print("Approximation difference:", full_prediction - linear_prediction, "N")

Here the changed input is 4.025 m/s. The tangent estimate is 8.82 N, while the full expression gives 8.81958 N. 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 = 5.0
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 speed at a hill crest inside the stated interval, predict whether required normal reaction 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: A negative required reaction means loss of contact, so the contact model cannot continue beyond that threshold. 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 N: 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(5.0) - evaluate(0.0))
for points in [101, 201, 401]:
    grid = np.linspace(0.0, 5.0, 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

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