Problem — Linear graph y = kx

Problem

Why is the law $y = kx$ such a recurring form in physics? Cite at least three concrete examples and explain how $k$ is derived from the graph.

Solution

Why it’s so recurring. The relation $y = kx$ expresses direct proportionality: doubling $x$ doubles $y$, and for $x=0$ we have $y=0$. It is the simplest form of non-trivial dependence, and very many physical laws, in the range where the effects are “small” or the idealisation holds, reduce exactly to a direct proportionality. The graph is a straight line through the origin, whose slope is the constant $k$: this makes it immediate to recognise and to measure.

Three (and more) concrete examples.

• Ohm’s law: $V = R\,I$ — the voltage is proportional to the current, with $k = R$ (resistance).
• Hooke’s law: $F = k\,x$ — the elastic force is proportional to the extension, with $k$ the spring’s elastic constant.
• Second law of dynamics (at fixed mass): $F = m\,a$ — at constant mass, the force is proportional to the acceleration, with $k = m$.
• Density: $m = \rho\,V$ — the mass is proportional to the volume, with $k = \rho$.

How to derive $k$ from the graph. Since the line passes through the origin, $k$ is its slope. Take two sufficiently distant points on the best-fit line, $(x_1,y_1)$ and $(x_2,y_2)$, and compute $k = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}.$ It is important to use two points of the interpolating line, not any two experimental data points (which carry scatter): the best-fit line averages out the random error. The dashed lines in the graph show how, having chosen a value of $x$, one reads off the corresponding $y = kx$ on the vertical axis.

$\ev{y=kx\ \Rightarrow\ \text{line through the origin};\quad k=\dfrac{\Delta y}{\Delta x}\ \text{(best-fit slope)}}$

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Topics: Metodo sperimentale Skills: Lettura dei grafici · Linearizzazione dei dati