idealized physical models and applicability limits

Physical theory connects observed quantities by laws, while experiment provides data for testing predictions. To apply theory, one builds an idealized model: essential variables are selected and assumptions are introduced, such as a point mass, an ideal gas, or a mathematical pendulum. Such a model predicts, for example, $T=2\pi\sqrt{L/g}$ for small pendulum oscillations or $pV=nRT$ for an ideal gas. Its limits of applicability are determined by how small the discarded factors are compared with the retained ones. If a prediction systematically disagrees with experiment, the model is refined: air resistance, body size, molecular interaction, or large angles are included. Therefore the same theory allows different models: the choice depends on the problem and the required accuracy.

For the model of vertical stone flight $h(t)=v_0 t-\frac{1}{2}gt^2$, name two assumptions and one condition under which the model is applicable.

A ball of mass $m=0{,}145$ kg and radius $r=0{,}050$ m moves at speed $v$. The model without air resistance is considered applicable while the drag force $F_d=\frac12 C\rho A v^2$ does not exceed $10\%$ of the weight. For $C=0{,}47$ and $\rho=1{,}2$ kg/m$^3$, estimate the maximum speed $v$ at which the model is still acceptable.

A mathematical pendulum has length $L=1{,}0$ m. The formula $T_0=2\pi\sqrt{L/g}$ is derived for small angles. Estimate the relative correction $T\approx T_0(1+\theta_0^2/16)$ for $\theta_0=5^\circ$ and $\theta_0=40^\circ$. Draw a conclusion about the applicability limit of the small-oscillation model.

Self-study simulation plan. Open a projectile motion simulation, for example PhET “Projectile Motion”, and switch on the no-air-resistance mode. Launch the projectile at fixed $v_0$ and angle, and record the range. Then switch on air resistance and repeat with the same parameters; vary mass, diameter, and speed one parameter at a time. Make a table: parameter, range without air resistance, range with air resistance, relative discrepancy. State for which values of speed and size the no-air-resistance model gives a discrepancy below $5\%$, and for which it needs refinement. Do not perform real throws near people or glass; use only the simulation.

Show that an idealized model is not reality itself but a chosen scheme with assumptions and a region of applicability.

Diagram: on the left, a real object (stone, pendulum, gas), an arrow to an “assumptions” box (point mass, no friction, small angles), then a “model equations” box and a “predictions” box. On the right, draw a parameter axis, for example speed or angle, and a shaded “model applicable” region; beyond its boundary show growing error and an arrow to a refined model. If useful, add a mini-graph of $|\text{error}|$ versus the parameter.

A ball of mass $0{,}43\ \text{kg}$ is kicked at $25\ \text{m/s}$ at $30^\circ$ to the horizontal. Without air resistance the model gives range $R=\frac{v^2\sin 2\theta}{g}=\frac{(25)^2\sin 60^\circ}{9{,}81}\approx 55\ \text{m}$. On a real pitch the ball may travel noticeably less at such a kick, especially if it is spinning.

This tests the idealization 'point mass + no air resistance'. Comparison with real flight shows the applicability limit: at high speeds and with spin, drag and the Magnus effect become significant.

A car of mass $1200\ \text{kg}$ travels at $50\ \text{km/h}\approx 13{,}9\ \text{m/s}$. In a simple braking model with friction coefficient $\mu=0{,}7$, the braking distance is $s=\frac{v^2}{2\mu g}=\frac{(13{,}9)^2}{2\cdot 0{,}7\cdot 9{,}81}\approx 14\ \text{m}$. On wet ice with $\mu=0{,}1$ the same formula gives about $99\ \text{m}$.

The example uses a model where the car is a rigid body, friction is constant, and air resistance and driver reaction time are neglected. The sharp change in result when the surface changes shows which conditions lie within the model's applicability limits.

A steel bridge of length $1000\ \text{m}$ is heated by $30^\circ\text{C}$ in summer. With the linear expansion coefficient of steel $\alpha\approx 12\cdot 10^{-6}\ ^\circ\text{C}^{-1}$, the elongation is $\Delta L=\alpha L\Delta T\approx 12\cdot 10^{-6}\cdot 1000\cdot 30=0{,}36\ \text{m}$. That is why bridges have expansion gaps and movable supports.

If the bridge were treated as a perfectly rigid body, $\Delta L$ would be zero. The observed elongation shows the applicability limit of that idealization and requires a model that includes thermal expansion and deformation.