Physics 9702/21 — October/November 2016

Background Concept

For a sphere of diameter $d$, the volume is

$$V = \frac{\pi d^3}{6}$$

Mass is related to density and volume by

$$m = \rho V$$

Combining these gives the provided formula:

$$m = \frac{\pi d^3 \rho}{6}$$

Understanding the Question

You are given $m = 7.5\ \text{kg}$ and $\rho = 8100\ \text{kg m}^{-3}$ for a metal sphere. Part (i) asks you to calculate the diameter $d$ using the given expression.

Approach

1. Rearrange the given equation to make $d$ the subject.
2. Substitute $m$ and $\rho$ (already in SI units).
3. Take the cube root to obtain $d$.

Step-by-Step Reasoning

Start from

$$m = \frac{\pi d^3 \rho}{6}$$

Multiply both sides by $6$ and divide by $\pi \rho$:

$$d^3 = \frac{6m}{\pi \rho}$$

Now take the power $1/3$ (cube root):

$$d = \left(\frac{6m}{\pi \rho}\right)^{1/3}$$

Substitute values:

$$d = \left(\frac{6 \times 7.5}{\pi \times 8100}\right)^{1/3}$$

Compute inside the brackets:

$$\frac{6 \times 7.5}{\pi \times 8100} \approx 1.77 \times 10^{-3}$$

Cube root:

$$d \approx (1.77 \times 10^{-3})^{1/3} \approx 0.121\ \text{m}$$

Key Takeaways

• Use $m = \rho V$ and the volume of a sphere to relate mass, density, and diameter.
• Rearranging for a variable inside a cube requires taking a cube root.

Common Mistakes

• Rearranging incorrectly (e.g. forgetting the cube on $d$).
• Using radius instead of diameter.
• Mixing units (e.g. using $\rho$ in $\text{g cm}^{-3}$ without conversion).

Things to Be Careful About

• $d$ is in metres because $\rho$ is in $\text{kg m}^{-3}$ and $m$ is in $\text{kg}$.
• Keep brackets when taking the cube root: $d = (6m/(\pi\rho))^{1/3}$, not $6m/(\pi\rho)^{1/3}$.

Background Concept

When a quantity $Q$ depends on measured quantities through powers, you can propagate percentage (fractional) uncertainties using:

• For multiplication/division: fractional uncertainties add.
• For a power: fractional uncertainty is multiplied by the absolute value of the power.

So if

$$Q \propto A^n$$

then

$$\frac{\Delta Q}{Q} = |n|\frac{\Delta A}{A}$$

Understanding the Question

You already found $d$ in (i). Now you must use the given percentage uncertainties in $m$ ($\pm 4\%$) and $\rho$ ($\pm 5\%$) to find the absolute uncertainty in $d$, and then present $d \pm \Delta d$ with an appropriate number of significant figures.

Approach

1. Rewrite $d$ to show how it depends on $m$ and $\rho$ (ignore constants like $6$ and $\pi$ for uncertainty).
2. Add percentage uncertainties for the ratio $m/\rho$.
3. Multiply by $1/3$ because of the cube root.
4. Convert the resulting percentage uncertainty in $d$ into an absolute uncertainty using $\Delta d = (\Delta d/d)\,d$.
5. Round the uncertainty to 1 significant figure (typical rule) and round $d$ to the same decimal place.

Step-by-Step Reasoning

From

$$d = \left(\frac{6m}{\pi\rho}\right)^{1/3}$$

constants $6$ and $\pi$ have no uncertainty here, so for uncertainty purposes:

$$d \propto \left(\frac{m}{\rho}\right)^{1/3}$$

1. Combine uncertainties in $m/\rho$ (division means add percentage uncertainties):

$$\%\Delta\left(\frac{m}{\rho}\right) = 4\% + 5\% = 9\%$$

1. Apply the power $1/3$ (cube root):

$$\%\Delta d = \frac{1}{3}\times 9\% = 3\%$$

1. Convert to absolute uncertainty using your value from (i), $d \approx 0.121\ \text{m}$:

$$\Delta d = 0.03 \times 0.121 = 3.63 \times 10^{-3}\ \text{m}$$

Round uncertainty to 1 s.f:

$$\Delta d \approx 0.004\ \text{m}$$

Then quote $d$ to the same decimal place (thousandths):

$$d = 0.121\ \text{m}$$

So

$$d = 0.121 \pm 0.004\ \text{m}$$

Key Takeaways

• For $Q = A^aB^b$, fractional uncertainties add: $\Delta Q/Q = |a|\Delta A/A + |b|\Delta B/B$.
• A cube root reduces the percentage uncertainty by a factor of 3.
• Quote uncertainties to 1 significant figure and match the decimal place of the quoted value.

Common Mistakes

• Using $5\% - 4\%$ instead of adding (percentage uncertainties add for multiplication/division).
• Forgetting to multiply by $1/3$ for the cube root.
• Giving a percentage uncertainty as an absolute uncertainty (or vice versa).
• Rounding $d$ to too few significant figures so it no longer matches the stated uncertainty.

Things to Be Careful About

• Percentage (fractional) uncertainty rules assume uncertainties are small and independent (the exam expects this method).
• Keep $d$ and $\Delta d$ to consistent decimal places: e.g. $0.121 \pm 0.004$, not $0.121 \pm 0.0036$ (over-precise) and not $0.12 \pm 0.004$ (mismatched precision).