Estimating g with a marble

Problem

In the lab you drop a marble from a height of $1{,}5\;\mathrm{m}$ and measure with a photogate timer a fall time $t = 0{,}553\;\mathrm{s}$. Estimate the experimental value of $g$ and compare it with the standard value, discussing the possible sources of error.

Solution

Model. The marble starts from rest and falls freely: the motion is uniformly accelerated with zero initial velocity. The equation of motion for the fall is therefore:

$h = \tfrac{1}{2}\,g\,t^2$

Finding $g$. Isolating the acceleration due to gravity:

$g = \frac{2h}{t^2} = \frac{2 \cdot 1{,}5}{0{,}553^2} = \frac{3{,}0}{0{,}3058} \approx 9{,}81\;\mathrm{m/s}^2$

Comparison. The value obtained coincides, within experimental uncertainty, with the standard value $g = 9{,}81\;\mathrm{m/s}^2$: the measurement is excellent.

Sources of error. The main ones are:

• uncertainty on the height $h$: the exact starting and ending position of the marble is hard to fix to the millimetre;
• reaction time of the photogates and their alignment, which introduce a small systematic error on the time;
• air resistance, which slightly slows the fall and tends to underestimate $g$ (an effect that is nonetheless negligible for a dense marble over just $1{,}5\;\mathrm{m}$);
• any non-zero initial velocity if the marble is not released gently.

Since $g$ depends on the square of the time, a small error on $t$ is amplified: it is worth repeating the measurement several times and averaging.

$\ev{g \approx 9{,}81\;\mathrm{m/s}^2}$

Collegamenti

Argomenti: Cinematica Concetti: Caduta libera Competenze: Stime di Fermi