Physics 9702/22 — May/June 2012

Background Concept

The equation relates the volume flow rate $\left(\frac{V}{t}\right)$ to the pressure difference $P$ and the pipe dimensions. If you need to find an unknown constant (here $C$), you rearrange algebraically to make it the subject.

A key practical skill is to convert all quantities into SI units before substituting, especially when powers are involved (e.g. $r^4$). A small change in $r$ has a large effect because it is raised to the 4th power.

Understanding the Question

You are given

$$\frac{V}{t} = \frac{\pi P r^4}{8 C l}$$

and a single set of measured values for $\frac{V}{t}$, $P$, $r$ and $l$. You must calculate $C$ numerically and give it in units of $\text{N s m}^{-2}$.

Approach

1. Rearrange the formula to make $C$ the subject.
2. Convert $r$ from mm to m.
3. Substitute the values and evaluate carefully (especially $r^4$).
4. Present the final value with a sensible number of significant figures.

Step-by-Step Reasoning

Start with

$$\frac{V}{t} = \frac{\pi P r^4}{8 C l}$$

Multiply both sides by $8Cl$ and divide by $\left(\frac{V}{t}\right)$:

$$C = \frac{\pi P r^4}{8l\left(\frac{V}{t}\right)}$$

Convert to SI:

• $\frac{V}{t} = 1.20 \times 10^{-6}\ \text{m}^3\ \text{s}^{-1}$
• $P = 2.50 \times 10^{3}\ \text{N m}^{-2}$
• $r = 0.75\ \text{mm} = 0.75 \times 10^{-3}\ \text{m}$
• $l = 0.250\ \text{m}$

Compute:

$$C = \frac{\pi (2.50 \times 10^{3}) (0.75 \times 10^{-3})^{4}}{8(0.250)(1.20 \times 10^{-6})}$$

Evaluating gives:

$$C \approx 1.04 \times 10^{-3}\ \text{N s m}^{-2}$$

Key Takeaways

• Rearrange expressions cleanly to isolate the required variable.
• Always convert to SI units before substituting.
• Be extra careful when a measured quantity is raised to a power (here $r^4$).

Common Mistakes

• Forgetting to convert $r$ from mm to m (gives an error factor of $10^{12}$ because of $r^4$).
• Misplacing brackets when rearranging, e.g. dividing by $8l$ but not by $\left(\frac{V}{t}\right)$.
• Arithmetic errors when calculating $(0.75 \times 10^{-3})^4$.

Things to Be Careful About

• Keep $\left(\frac{V}{t}\right)$ together as a single quantity in the denominator.
• Use standard form to reduce power-of-ten mistakes.
• Carry a few extra digits during the calculation, then round at the end.

Background Concept

When a result is calculated from several measured quantities, its uncertainty comes from the uncertainties in those measurements.

For multiplication/division, you add fractional uncertainties:

$$Q = \frac{ab}{c} \Rightarrow \frac{\Delta Q}{Q} \approx \frac{\Delta a}{a} + \frac{\Delta b}{b} + \frac{\Delta c}{c}$$

For a power law, the fractional uncertainty is multiplied by the power:

$$Q = x^n \Rightarrow \frac{\Delta Q}{Q} \approx n\frac{\Delta x}{x}$$

These are the standard Cambridge A-Level rules used in data handling.

Understanding the Question

You already found $C$ in part (a). Now you must use the given measurement uncertainties to determine the absolute uncertainty in $C$ (in $\text{N s m}^{-2}$).

The key feature is that $r$ is raised to the 4th power, so its uncertainty contributes strongly.

Approach

1. Write $C$ in proportional form to see the dependence on each variable.
2. Compute each fractional uncertainty: $\Delta x/x$.
3. Multiply the radius fractional uncertainty by 4 because of $r^4$.
4. Add the fractional uncertainties.
5. Convert to absolute uncertainty using $\Delta C = C \times \left(\Delta C/C\right)$.

Step-by-Step Reasoning

From

$$C = \frac{\pi P r^4}{8l\left(\frac{V}{t}\right)}$$

constants ($\pi$, $8$) do not contribute uncertainty, so

$$C \propto \frac{Pr^4}{l\left(\frac{V}{t}\right)}$$

Now calculate fractional uncertainties from the table:

Pressure:

$$\frac{\Delta P}{P} = \frac{0.05}{2.50} = 0.020 = 2.0\%$$

Radius (note the factor 4):

$$\frac{\Delta r}{r} = \frac{0.01}{0.75} = 0.0133$$ $$4\frac{\Delta r}{r} = 4(0.0133) = 0.0533 = 5.33\%$$

Length:

$$\frac{\Delta l}{l} = \frac{0.001}{0.250} = 0.0040 = 0.40\%$$

Flow rate:

$$\frac{\Delta (V/t)}{(V/t)} = \frac{0.01}{1.20} = 0.00833 = 0.833\%$$

Add them (because they are multiplied/divided):

$$\frac{\Delta C}{C} = 0.020 + 0.0533 + 0.0040 + 0.00833 = 0.0857$$

So the percentage uncertainty is about $8.6\%$.

Convert to an absolute uncertainty using the calculated value $C = 1.04 \times 10^{-3}\ \text{N s m}^{-2}$:

$$\Delta C = (0.0857)(1.04 \times 10^{-3}) = 8.9 \times 10^{-5}\ \text{N s m}^{-2}$$

Key Takeaways

• For products/quotients: add fractional uncertainties.
• For powers: multiply fractional uncertainty by the power.
• A variable raised to a high power (like $r^4$) often dominates the final uncertainty.

Common Mistakes

• Forgetting the factor of 4 for $r^4$.
• Subtracting fractional uncertainties because a quantity is in the denominator (you still add).
• Using absolute uncertainties directly without first converting to fractional/percentage.
• Rounding fractional uncertainties too early, causing noticeable error in the final $\Delta C$.

Things to Be Careful About

• Use consistent significant figures during intermediate steps; round only at the end.
• Ensure you use the correct uncertainty for each quantity (e.g. $0.05$ in units of $10^3\ \text{N m}^{-2}$ is already built into the table entry).
• State the final uncertainty with the correct unit ($\text{N s m}^{-2}$).

Background Concept

Experimental results should be quoted with an uncertainty, and both should be rounded sensibly:

• The uncertainty is usually quoted to 1 significant figure (sometimes 2 if the first digit is 1 or 2).
• The measured/calculated value should then be rounded to the same decimal place as the uncertainty.

This ensures you do not claim more precision than your uncertainty allows.

Understanding the Question

You have a calculated value of $C$ (from part (a)) and an absolute uncertainty $\Delta C$ (from part (b)). You must present $C \pm \Delta C$ to an appropriate number of significant figures.

Approach

1. Round $\Delta C$ to 1 significant figure (unless it begins with 1 or 2).
2. Round $C$ to match the place value of $\Delta C$.
3. Present as either standard form or ordinary decimal form, with units.

Step-by-Step Reasoning

From earlier parts:

$$C = 1.04 \times 10^{-3}\ \text{N s m}^{-2}$$ $$\Delta C = 8.9 \times 10^{-5}\ \text{N s m}^{-2}$$

Round the uncertainty to 1 s.f. (first digit is 8):

$$\Delta C \approx 9 \times 10^{-5}\ \text{N s m}^{-2}$$

It can be helpful to express this with the same power of ten as $C$:

$$9 \times 10^{-5} = 0.09 \times 10^{-3}$$

So the final quoted result is:

$$C = (1.04 \pm 0.09) \times 10^{-3}\ \text{N s m}^{-2}$$

(Equivalently $C = 0.00104 \pm 0.00009\ \text{N s m}^{-2}$.)

Key Takeaways

• Quote uncertainty to 1 s.f. (typical A-Level convention).
• Quote the value to the same precision implied by the uncertainty.
• Standard form makes the matching of significant figures clearer.

Common Mistakes

• Quoting too many significant figures in $C$, e.g. $1.035 \times 10^{-3}$ when the uncertainty is about $10^{-4}$.
• Rounding the uncertainty to too many figures (e.g. keeping $8.9 \times 10^{-5}$) when only 1 s.f. is expected.
• Rounding the value but forgetting to round the uncertainty, or vice versa.

Things to Be Careful About

• If the uncertainty had started with 1 or 2 (e.g. $1.2 \times 10^{-4}$), quoting it to 2 s.f. is often acceptable; here it starts with 8, so 1 s.f. is appropriate.
• Ensure both value and uncertainty have consistent powers of ten if using standard form.
• Always include the unit $\text{N s m}^{-2}$ in the final statement.