Buoyancy models

Buoyancy is the net pressure force on a body in a fluid. Pressure is larger on deeper parts of the surface, so a fluid in a downward gravitational field produces a net upward force. In a uniform-density fluid, that force equals the weight of the displaced fluid.

Archimedes' principle

For fluid density \(\rho_{\mathrm f}\), displaced volume \(V_{\mathrm d}\), and gravitational acceleration magnitude \(g\), buoyant-force magnitude is

\[F_{\mathrm B}=\rho_{\mathrm f}gV_{\mathrm d}\]

The direction is opposite gravity. In vector form,

\[\mathbf F_{\mathrm B}=-\rho_{\mathrm f}V_{\mathrm d}\mathbf g\]

where \(\mathbf g\) points downward. The displaced volume is the part of the body's volume below the fluid surface, not necessarily its full volume.

import numpy as np

fluid_density, volume = 1000.0, 0.002
gravity_vector = np.array([0.0, -9.81])
buoyant_force = -fluid_density * volume * gravity_vector
print("Fully immersed buoyancy (N):", buoyant_force)

The force acts through the center of buoyancy, the centroid of displaced fluid for a uniform density. That location matters for rotational stability.

Why displaced-fluid weight appears

Imagine replacing the submerged body with fluid identical to its surroundings. That imagined parcel would be in hydrostatic equilibrium: the net pressure force on its boundary would balance its weight. Restoring the body leaves the same external pressure distribution when it does not otherwise disturb the static fluid, so the net pressure force equals the weight of the replaced fluid.

Equivalently, integrate pressure over the body surface:

\[\mathbf F_{\mathrm B}=-\oint_A p\,\hat{\mathbf n}\,dA\]

With hydrostatic relation \(\boldsymbol\nabla p=\rho_{\mathrm f}\mathbf g\), the surface integral reduces to the displaced-fluid result. Uniform atmospheric pressure on a closed surface cancels; only the pressure gradient creates net buoyancy.

Floating, sinking, and neutral buoyancy

Object weight is

\[\mathbf W=m\mathbf g\]

For static vertical balance without other support,

\[\rho_{\mathrm f}gV_{\mathrm d}=mg\]

so the required displaced volume is

\[V_{\mathrm d}=\frac{m}{\rho_{\mathrm f}}\]

If a homogeneous object of total volume \(V\) floats in a uniform fluid, its immersed volume fraction is

\[\frac{V_{\mathrm d}}{V}=\frac{\rho_{\mathrm o}}{\rho_{\mathrm f}}\]

where \(\rho_{\mathrm o}=m/V\). An average object density below the fluid density permits partial floating. Equal densities give neutral buoyancy when fully submerged. Greater average density produces downward net force unless another force or trapped lower-density volume changes the system.

A steel ship floats because its total mass divided by its enclosed external volume can be less than water density. Material density alone is not the relevant average.

Apparent weight

A fully submerged object supported by a scale or string has reduced apparent weight. If it is stationary,

\[T=mg-\rho_{\mathrm f}gV\]

where \(T\) is the scale or string force. The object's gravitational weight has not changed; the fluid supplies an additional upward force. Calling buoyancy “reduced gravity” can hide this force balance.

Partial immersion is a geometry problem

For a vertical object with known cross-sectional area as a function of height, displaced volume is an integral up to the free surface. For a constant-area prism,

\[V_{\mathrm d}=Ah\]

until fully submerged. Buoyancy then grows linearly with immersion depth \(h\). Curved bodies have nonlinear volume-depth relations.

area, body_height, bottom_height = 0.01, 0.2, -0.12
immersed_height = np.clip(-bottom_height, 0.0, body_height)
displaced_volume = area * immersed_height
print(f"Displaced volume: {displaced_volume:.6f} m^3")

Near the waterline, inaccurate clipping can create discontinuous force and artificial jitter. Refine geometry or use a smooth, volume-consistent integration whose error is measured.

Translational motion

With upward coordinate positive, a simple vertical model may be written

\[m\ddot y=F_{\mathrm B}(y)-mg+F_{\mathrm{drag}}(y,\dot y)+F_{\mathrm{contact}}\]

Buoyancy depends mainly on displaced volume and fluid density. Drag depends on relative velocity, shape, flow regime, and viscosity. Combining them into one arbitrary upward force prevents independent validation.

mass, upward_velocity, drag_coefficient = 1.0, 0.2, 2.0
buoyancy = fluid_density * displaced_volume * 9.81
weight = -mass * 9.81
drag = -drag_coefficient * upward_velocity
acceleration = (buoyancy + weight + drag) / mass
print(f"Vertical acceleration: {acceleration:.3f} m/s^2")

An object displaced slightly downward from a stable floating equilibrium displaces more fluid and gains upward restoring force. Without damping it can oscillate; drag dissipates mechanical energy and allows settling. Fluid acceleration around the object can add effective inertia called added mass, which this elementary model omits.

Orientation and rotational stability

Weight acts through the center of mass, while buoyancy acts through the center of buoyancy. Their separation can produce torque:

\[\boldsymbol\tau=(\mathbf r_{\mathrm B}-\mathbf r_{\mathrm{cm}})\times\mathbf F_{\mathrm B}\]

Tilting changes the immersed shape and moves the center of buoyancy. A floating body's orientation is stable when a small tilt creates a restoring torque under the stated load and geometry. Center-of-mass height alone is not enough; hull waterplane geometry and the shift of buoyancy matter. A purely vertical point-mass model cannot predict capsizing.

Varying fluid density

If density changes with depth, buoyant force is the weight of the displaced fluid with its actual density distribution:

\[\mathbf F_{\mathrm B}=-\left(\int_{V_{\mathrm d}}\rho(\mathbf x)\,dV\right)\mathbf g\]

The force acts through a density-weighted center, not necessarily the geometric centroid. An object can settle at a depth where its average density matches surrounding stratification, though stability and mixing require more analysis.

Gas buoyancy follows the same principle. A balloon rises when displaced air weighs more than the balloon system, including envelope and payload. Air density varies with altitude, so a constant-density model eventually fails.

Boundaries and missing effects

Contact with the bottom adds a normal force and can reduce fluid access beneath an object, altering pressure distribution. Surface tension matters for small objects and contact lines. Waves, splashing, trapped air, fluid entry, porous flow, leakage, and changing mass need richer models.

A static Archimedes calculation assumes a hydrostatic surrounding fluid. Strong acceleration, rotation, or flow can make pressure non-hydrostatic. Then integrate pressure or solve fluid motion instead of adding a universal displaced-volume force blindly.

Verification checks

Zero displaced volume must give zero buoyancy. Doubling displaced volume in a uniform fluid must double force. A fully submerged object whose average density equals fluid density should have buoyancy equal and opposite to weight. Check force units and signs.

For partial immersion, compare numerical displaced volume with analytic shapes such as a box or sphere. Reduce geometry and time resolution independently, monitor force and torque continuity at the surface, and verify that any energy loss comes from an explicit drag, collision, or wave model.

Buoyancy is best understood as a pressure result. Reliable computation therefore begins with geometry and hydrostatics, not a decorative upward arrow.

A worked Python notebook

A vertical rectangular block has horizontal cross-sectional area 0.01 square metres. The question for this calculation is precise: how does buoyant force magnitude change when we change immersed height, 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 immersed height cannot exceed the block's 0.2 metre height. 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 0.2 m. 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 immersed height, expressed in m. The output is buoyant force magnitude, 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.

\[f(q) = 98.1 q\]

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 0.12 m, the expression gives 11.772 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("1000*9.81*0.01*q", locals={"q": q})
reference_input = 0.12
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "m")
print("Predicted buoyant force magnitude:", 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 0 N. The midpoint, at 0.1 m, gives 9.81 N. The upper endpoint gives 19.62 N. Read these together with the assumptions: they describe different choices of immersed height 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, 0.1, 0.2])
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 -> {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 immersed height; the vertical axis shows buoyant force magnitude. 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, 0.2, 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("Immersed height (m)")
ax.set_ylabel("Buoyant force magnitude (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 buoyant force magnitude change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 0.12 m is 98.1 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.

\[f'(q) = 98.1\]

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 = 0.121
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 0.121 m. The tangent estimate is 11.8701 N, while the full expression gives 11.8701 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 = 0.2
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 immersed height inside the stated interval, predict whether buoyant force magnitude 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 immersed height cannot exceed the block's 0.2 metre height. 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(0.2) - evaluate(0.0))
for points in [101, 201, 401]:
    grid = np.linspace(0.0, 0.2, 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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