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Power and energy flow

Power is the rate at which energy crosses a system boundary or changes form. It answers “how fast?” rather than “how much?” A large energy transfer over a long time can have modest power, while a brief transfer can have high power even when its total energy is small.

Average and instantaneous power

Average power over a nonzero interval is

\[P_{\mathrm{avg}}=\frac{\Delta E}{\Delta t}\]

Instantaneous power is

\[P=\frac{dE}{dt}\]

The SI unit is the watt:

\[1\,\mathrm W=1\,\mathrm{J/s}\]

A power reading is tied to an instant or interval. Dividing by a frame count instead of physical elapsed time gives a display-dependent quantity, not power.

Mechanical power from a force

For a force acting at a point moving with velocity \(\mathbf v\),

\[P=\mathbf F\cdot\mathbf v\]

Only the force component parallel to velocity transfers energy instantaneously. Power is positive when the force adds kinetic energy to the selected particle, negative when it removes kinetic energy, and zero when force is perpendicular to velocity.

For a constant horizontal force \(F=200\,\mathrm N\) acting along velocity \(v=3.0\,\mathrm{m/s}\),

\[P=Fv=600\,\mathrm W\]

If the same force is perpendicular to motion, its power on the particle is zero even though it can change velocity direction.

import numpy as np

force_n = np.array([6.0, 0.0])
velocity_m_per_s = np.array([2.0, 1.0])
power_w = np.dot(force_n, velocity_m_per_s)
print(f"Mechanical power: {power_w:.1f} W")

For an extended body, use the velocity of the force application point, not automatically the center-of-mass velocity.

Rotational power

For rotation about a fixed axis, torque and angular velocity give

\[P=\boldsymbol\tau\cdot\boldsymbol\omega\]

Torque in newton-metres multiplied by angular speed in radians per second yields watts. Radians are dimensionless in SI, but the angular convention still matters. A positive dot product adds rotational kinetic energy.

Translational

\(\mathbf F\cdot\mathbf V_{\mathrm{CM}}\)

and rotational

\(\boldsymbol\tau_{\mathrm{CM}}\cdot\boldsymbol\omega\)

terms can both contribute to rigid-body power. Counting the application-point power and then adding both decomposed terms would double-count the same transfer.

Power as the derivative of work

Since differential work is

\(dW=\mathbf F\cdot d\mathbf r\)

and \(d\mathbf r/dt=\mathbf v\),

\[\frac{dW}{dt}=\mathbf F\cdot\frac{d\mathbf r}{dt}=\mathbf F\cdot\mathbf v\]

Integrating power recovers work:

\[W=\int_{t_1}^{t_2}P(t)\,dt\]

This creates a two-way data check: integrate measured power and compare with independently calculated energy change over the same interval.

A system energy ledger

Choose positive transfer into the system. A general rate balance can be written

\[\frac{dE_{\mathrm{system}}}{dt}=\sum P_{\mathrm{in}}-\sum P_{\mathrm{out}}\]

Alternatively, store every signed channel with one convention and sum it once. Mixing a signed convention with separate input and output lists can reverse or double a flow.

channels_w = {"heater in": 100.0, "heat loss out": -30.0, "shaft work out": -20.0}
stored_energy_rate = sum(channels_w.values())
print(f"Rate of stored energy increase: {stored_energy_rate:.1f} W")

The boundary controls classification. Work by friction can be an energy transfer into a block-only system, while frictional heating is an internal conversion for a system that contains both contacting surfaces.

Multiple energy stores

A mechanical system might have kinetic, gravitational, elastic, and thermal stores:

\[E_{\mathrm{system}}=K+U_g+U_s+E_{\mathrm{thermal}}\]

Their rates satisfy a ledger such as

\[\frac{dK}{dt}+\frac{dU_g}{dt}+\frac{dU_s}{dt}+\frac{dE_{\mathrm{thermal}}}{dt}=P_{\mathrm{external}}\]

One store can decrease while total system energy stays constant. During ideal spring oscillation, elastic and kinetic energies exchange. A plot of only kinetic energy can look like a loss even when the sum is constant.

Conservative-force power

For a time-independent potential with \(\mathbf F=-\nabla U\),

\[P_{\mathrm{cons}}=\mathbf F\cdot\mathbf v=-\frac{dU}{dt}\]

Thus positive work by the conservative force reduces potential energy by the same rate. If the potential explicitly depends on time, the total derivative also contains \(\partial U/\partial t\), so this simple exchange statement needs modification.

Dissipation and drag

For linear drag in a stationary fluid, \(\mathbf F_D=-b\mathbf v\),

\[P_D=\mathbf F_D\cdot\mathbf v=-bv^2\le0\]

The body's mechanical energy decreases. In a larger closed system, energy appears in fluid motion and internal thermal energy. A negative mechanical-power channel should not be described as energy destruction.

With moving fluid, drag is based on relative velocity. The power delivered to the body uses body velocity in the chosen frame, so a flowing fluid can do positive work on it.

Efficiency

Efficiency compares useful output energy or power with supplied input:

\[\eta=\frac{P_{\mathrm{useful,out}}}{P_{\mathrm{input}}}\]

For a passive device over a consistent steady interval, \(0\le\eta\le1\). Values above one usually indicate mismatched boundaries, stored energy being released, sign mistakes, or different averaging intervals—not creation of energy.

Instantaneous efficiency can be misleading when input and output are out of phase or energy is being stored. Compare integrated energies over a complete operating interval or use steady-state average powers.

Power from sampled energy data

For timestamped energy samples, a centered estimate at an interior time is

\[P(t_i)\approx\frac{E(t_{i+1})-E(t_{i-1})}{t_{i+1}-t_{i-1}}\] 
times_s = np.linspace(0, 4, 41)
energies_j = 5.0 + 3.0 * times_s**2
power_estimates_w = np.gradient(energies_j, times_s, edge_order=2)
print(f"Power at 2 s: {power_estimates_w[20]:.2f} W")

Differentiation amplifies measurement noise. Smoothing changes the estimate and can hide short power peaks, so report the method and window. Averages over intervals are often more robust than claims about instantaneous power.

Energy from sampled power

Trapezoidal integration uses

\[\Delta E\approx\sum_i\frac{P_i+P_{i+1}}{2}(t_{i+1}-t_i)\] 
energy_transfer_j = np.trapezoid(power_estimates_w, times_s)
print(f"Integrated energy transfer: {energy_transfer_j:.2f} J")
assert np.isclose(energy_transfer_j, energies_j[-1] - energies_j[0])

Refine the time sampling or compare another quadrature rule when power varies rapidly. An impulsive event may require its impulse or event energy to be recorded separately rather than inferred from sparse continuous samples.

Peak, average, and rated power

Peak power, interval-average power, and long-term rated power answer different questions. A motor can briefly exceed its continuous rating by drawing on stored thermal capacity. A sensor's limited bandwidth can miss true peaks. Report the averaging window and sampling bandwidth beside a peak claim.

For periodic power, average over complete cycles after transients. Choosing a partial cycle can produce a phase-dependent average that does not represent steady operation.

Uncertainty

For mechanical power \(P=Fv\cos\theta\), uncertainty can arise from force, speed, alignment angle, timestamp synchronization, and frame choice. Force and velocity measurements may be correlated. Near perpendicular alignment, a small angle error can dominate a small true power.

Integrated energy uncertainty also depends on correlations between samples. Treating every high-rate sample as statistically independent can understate uncertainty when sensor errors drift slowly.

Validation checks

Power makes energy accounting dynamic. A trustworthy rate always names the system, channel, sign, time interval, and measurement or numerical procedure that produced it.

A worked Python notebook

A constant 6 newton force acts along an axis while the body's signed velocity varies. The question for this calculation is precise: how does mechanical power change when we change speed along the force, 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 force and velocity must refer to the same point of application and reference frame. 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 -3 to 3 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 along the force, expressed in m/s. The output is mechanical power, expressed in W. 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.

\[f(q) = 6 q\]

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 2 m/s, the expression gives 12 W. 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("6*q", locals={"q": q})
reference_input = 2.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "m/s")
print("Predicted mechanical power:", reference_output, "W")

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 -18 W. The midpoint, at 0 m/s, gives 0 W. The upper endpoint gives 18 W. Read these together with the assumptions: they describe different choices of speed along the force 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([-3.0, 0.0, 3.0])
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} W")
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 along the force; the vertical axis shows mechanical power. 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(-3.0, 3.0, 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 along the force (m/s)")
ax.set_ylabel("Mechanical power (W)")
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 mechanical power change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 2 m/s is 6 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.

\[f'(q) = 6\]

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 = 2.03
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, "W")
print("Full expression:", full_prediction, "W")
print("Approximation difference:", full_prediction - linear_prediction, "W")

Here the changed input is 2.03 m/s. The tangent estimate is 12.18 W, while the full expression gives 12.18 W. 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 = 6.0
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 along the force inside the stated interval, predict whether mechanical power 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 force and velocity must refer to the same point of application and reference frame. 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 W: 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(3.0) - evaluate(-3.0))
for points in [101, 201, 401]:
    grid = np.linspace(-3.0, 3.0, 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.

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