INPhO 2022, Теория — Задача 4

• (a) The student recorded only the length of the string and did not add the radius of the bob, so l (and hence g) was underestimated.
• (b) Omitting the bob radius only adds a constant intercept b = 4π²r/g to T² = al + b; the slope a = 4π²/g is unaffected.

(a) A low value of $g$ from the formula $g = 4\pi^2 l/T^2$ can result either from an underestimation of $l$ or an overestimation of $T$.

The latter can happen in case of either a fast-running clock (stopwatch) or a consistent error in the measurement. We are told that the instruments were accurate and the measurements were properly made. So this possibility is ruled out.

Under the given assumption that the measurements of $l$ were accurate, the only way that the value of $l$ can be underestimated systematically is by ignoring the size of the bob of the pendulum. Since the student recorded only the length of the string, and did not add the radius of the bob, this caused an underestimation of $l$, and consequently, of $g$.

(b) When $T^2$ is plotted against $l$, one should obtain a linear graph of the form

where, in the ideal case, $l$ represents the distance of the centre of mass of the bob from the suspension point, the slope $a = \dfrac{4\pi^2}{g}$, and the intercept $b = 0$.

If the size of the bob is not included in the measurement of $l$, one should still get a linear graph, with same value of $a$, but now with $b = \dfrac{4\pi^2 r}{g}$, where $r$ is the radius of the bob. Careful inspection of the graph indeed shows a positive intercept on the $T^2$-axis, confirming this scenario. However, since the slope of the line is unaffected by the underestimation of $l$, the value of $g$ is still correctly determined from the slope.

Note that an overestimation of $T$ by $\Delta T$, say, would have caused the slope of the graph, and hence the derived $g$, to change since $T^2$ would have been modified by a term proportional to $T$ itself.