Physical phenomena and models

A physical phenomenon is a regularity of nature observed in experiment: free fall, heating, electrification. To describe a phenomenon quantitatively, physics builds a model — a simplified image of the object in which some properties are discarded and the rest are replaced by idealizations (point mass, ideal gas, harmonic oscillator). A model always introduces assumptions: for a falling stone one neglects air resistance and treats $g$ as constant. The model links quantities by equations, and its predictions are compared with measurements: for example, the distance travelled in uniformly accelerated motion from rest is

$$s=\frac{1}{2}at^2 .$$

Agreement within experimental uncertainty does not prove the model 'true'; it only confirms that the model is applicable under these conditions. Hence every model has a range of applicability: a point mass works when the body's size is small compared with the distance travelled, an ideal gas works at low density and far from condensation. Scientific description is a cycle: observation → idealization → equation → prediction → test → refinement.

Classify each item as a physical phenomenon (P) or a model (M): (a) free fall of a stone; (b) point mass; (c) ideal gas; (d) electrification of a comb by friction; (e) harmonic oscillator. For each model, name the property that was discarded.

A student measures the fall time of a steel ball from a height $h=2.0$ m and obtains $t_{exp}=0.66$ s (in seconds). Build the model 'point mass without air resistance', obtain the prediction $t_{mod}$, compute the relative deviation $\delta$, and list the model assumptions.

For a pendulum of length $L=1.00$ m the small-oscillation model gives $T_0=2\pi\sqrt{L/g}$. Measurements give $T=2.01$ s at amplitude $\theta_0=5^\circ$ and $T=2.09$ s at $\theta_0=45^\circ$. Compute $T_0$, find $\delta$ for both cases, check the correction $T\approx T_0\left(1+\frac{\theta_0^2}{16}\right)$ (angle in radians), and state the range of applicability of the small-oscillation model.

A safe self-study plan in a simulation (no real weights and no sharp objects).

1. Open any pendulum simulation (for example, Pendulum Lab) and fix $L=1.00$ m, the mass and $g$ at their default values.
2. For amplitudes $\theta_0=2^\circ,5^\circ,10^\circ,20^\circ,45^\circ$, measure the time of 5 full oscillations and find the period $T$ by dividing by 5.
3. Compare each measurement with the model $T_0=2\pi\sqrt{L/g}$ and compute $\delta$.
4. Plot $\delta(\theta_0)$ and mark the 1% level.
5. Record in your notebook: which assumptions the model contains, at which amplitude it leaves its range of applicability, and which correction must be introduced.
   All measurements are made in the simulation; carry out real pendulum measurements only with a fixed support and no risk of the load falling.

To show the cycle of building a scientific description and the range of applicability: how an observed phenomenon maps onto an idealized model, which assumptions are discarded, and how the deviation grows beyond the range of applicability.

Diagram (flowchart): 'Phenomenon → idealization (list of assumptions) → model equations → prediction → comparison with experiment → refinement of the model', with a feedback loop from the comparison back to the idealization. Next to it, a graph: the X axis is the amplitude $\theta_0$ from $0$ to $50^\circ$, the Y axis is the relative deviation $\delta$ in percent; the measured points lie on a curve growing roughly as $\theta_0^2$, and the horizontal dashed line at $1\%$ marks the range of applicability of the small-oscillation model (it crosses the curve near $23^\circ$).

A runner covers $100\ \text{m}$ in $10.5\ \text{s}$. To estimate the average speed, the coach divides distance by time: $v=\frac{100\ \text{m}}{10.5\ \text{s}}\approx 9.5\ \text{m/s}.$ The runner's height ($1.8\ \text{m}$), arm swing and body sway are not included in the calculation.

This is a direct test of replacing a real object with a point mass: the model keeps only the properties needed for the quantity $v=s/t$, and discards everything else as irrelevant to the stated task.

A phone of mass $0.20\ \text{kg}$ rests on a table and then falls from a height of $1.2\ \text{m}$. The sensor reads an acceleration near $9.8\ \text{m/s}^2$ in free fall and a support force $F\approx 2.0\ \text{N}$ at rest. In the calculation the phone is treated as a rigid body with the sensor at a single point.

This checks that a model can be matched against measurements: the prediction $F=mg=0.20\ \text{kg}\cdot 9.8\ \text{m/s}^2\approx 2.0\ \text{N}$ is compared with the sensor reading, and any discrepancy marks the limits of the model's validity.

A bridge span of length $L=1200\ \text{m}$ carries a convoy of trucks with a total mass $m=3.0\cdot 10^{5}\ \text{kg}$. The engineer computes the weight of the load $P=mg=3.0\cdot 10^{5}\ \text{kg}\cdot 9.8\ \text{m/s}^2\approx 2.9\ \text{MN}$ and distributes it along the length. The shape of the span, wind gusts of $15\ \text{m/s}$ and a temperature of $+30\ ^\circ\text{C}$ are left out of this estimate and checked with separate models.

The example shows a complex phenomenon split into separate models: one describes only the weight of the load, another thermal expansion at $+30\ ^\circ\text{C}$, a third the wind; together they cover what a single model cannot.