Problem — Which graph do you expect (T vs m)

Problem

You measure the period $T$ of a pendulum for five different masses ($m = 50,\ 100,\ 200,\ 500,\ 1000$ g) while keeping $L$ and the amplitude constant. Knowing that the period does not depend on the mass, predict qualitatively the graph of $T$ vs $m$. Draw it; state how you would recognise, from the data, a small spurious dependence caused by air resistance.

Solution

Prediction. The period of a pendulum (for small oscillations) is $T = 2\pi\sqrt{\frac{L}{g}}$ and does not contain the mass $m$. Since $L$, $g$ and the amplitude are held constant, $T$ must be constant as $m$ varies.

In the $T$ vs $m$ plane we therefore expect a horizontal straight line: all experimental points aligned at the same height $T_0$, regardless of the value of $m$. This is exactly the scheme drawn above.

How to recognise a spurious dependence. Air resistance introduces a small dissipative force that weighs more heavily on light masses (for equal volume, a small mass has less inertia to overcome relative to the drag). If this effect were present, the points would not lie perfectly on a horizontal line: a slight systematic slope or drift would be observed, for example a slightly larger (or smaller) $T$ for the smaller masses, tending towards the ideal value $T_0$ as $m$ increases.

In practice the warning sign is: the points show a regular, monotonic trend (not random scatter around the horizontal) correlated with $m$. Random scatter would just be measurement error; a systematic slope instead indicates a residual physical effect such as air resistance.

$\ev{T(m)=\text{costante}=T_0\ \Rightarrow\ \text{retta orizzontale; deriva sistematica}\Rightarrow\text{effetto spurio}}$

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Topics: Metodo sperimentale Skills: Lettura dei grafici