A student measured a metal ring of the type shown below - Edexcel - A-Level Physics - Question 4 - 2023 - Paper 6

To calculate the mean value of $d$, sum the individual measurements and divide by the number of measurements:

Mean: $d = \frac{8.53 + 8.56 + 8.55 + 8.53}{4} = \frac{34.17}{4} = 8.54$ mm.

To find the uncertainty, we consider the maximum deviation from the mean value:

Uncertainty: $U(d) = 0.02$ mm (taking the furthest reading from the mean into account).

To find the uncertainty in $d^{2}$, use the formula:

$U(d^{2}) = d^{2} \times \left( \frac{U(d)}{d} \right) \times 2$

Substituting the values, we have:

• $d = 10.70$ mm
• $U(d) = 0.06$ mm

$U(d^{2}) = (10.70)^{2} \times \left( \frac{0.06}{10.70} \right) \times 2 \approx 1$ mm$^{2}$.

To find the percentage uncertainty in the area $A$, we can use the formula:

$\text{Percentage Uncertainty} = \left( \frac{U(A)}{A} \right) \times 100\%$

Where $U(A) \approx 2 \times U(d^{2}) = 0.4\%$.

Measuring the total mass of 10 metal rings improves accuracy due to averaging over multiple samples, minimizing the impact of random error. This provides a more reliable value compared to measuring a single ring, which may have larger uncertainties.

To calculate the mean density $\rho$, we use the formula:

$\rho = \frac{m_{0}}{V}$

Where:

• $m_{0} = 63.0 g \pm 0.5 g$
• $x_{0} = 14.03 mm \pm 0.04 mm$
• To convert to cm$^{3}$, note that the total volume $V$ can be calculated based on thickness and area, leading to\n\n a computed value for density.

Given that the density of stainless steel ranges from 7.48 g cm$^{-3}$ to 7.95 g cm$^{-3}$, if the computed density falls within this range, we can conclude that the metal rings could indeed be made from stainless steel.