Physics 9702/22 — February/March 2020

Background Concept

When a quantity is calculated from measured variables using a formula, you substitute the measured values into the equation and evaluate it. Here,

$$g = \frac{4\pi^2 l}{T^2}$$

where $l$ is the pendulum length (in m) and $T$ is the period (in s). The unit check is useful: $l$ has unit m and $T^2$ has unit $\text{s}^2$, so $g$ has unit $\text{m s}^{-2}$.

Understanding the Question

You are given $l = 1.50\ \text{m}$ and $T = 2.48\ \text{s}$ and asked to calculate $g$ using the provided equation.

Approach

1. Square the period $T$.
2. Calculate $4\pi^2 l$.
3. Divide to get $g$.
4. Quote the answer with unit $\text{m s}^{-2}$.

Step-by-Step Reasoning

Start with

$$g = \frac{4\pi^2 l}{T^2}$$

Calculate the denominator:

$$T^2 = (2.48)^2 = 6.1504$$

Calculate the numerator:

$$4\pi^2 l = 4\pi^2 \times 1.50$$

Using $4\pi^2 \approx 39.48$:

$$4\pi^2 l \approx 39.48 \times 1.50 = 59.22$$

Now divide:

$$g = \frac{59.22}{6.1504} = 9.63\ \text{m s}^{-2}$$

Key Takeaways

• Substitute into the given equation carefully.
• Square only the $T$ term (not the whole fraction).
• Always include the unit and sensible significant figures.

Common Mistakes

• Forgetting to square $T$.
• Squaring $2.48$ incorrectly (calculator error).
• Missing the unit or using $\text{m s}^{-1}$ instead of $\text{m s}^{-2}$.

Things to Be Careful About

• Keep enough calculator precision until the final step, then round appropriately.
• The uncertainty information given is not needed for part (i); it is used in parts (ii) and (iii).

Background Concept

For uncertainties in a calculated quantity:

• When quantities are multiplied or divided, their percentage (or fractional) uncertainties add.
• When a quantity is raised to a power $n$, its percentage uncertainty is multiplied by $n$.

So if

$$y = \frac{a\,b}{c^2}$$

then

$$\%\Delta y = \%\Delta a + \%\Delta b + 2(\%\Delta c)$$

Understanding the Question

You are given percentage uncertainties in $l$ and $T$, and you must find the percentage uncertainty in

$$g = \frac{4\pi^2 l}{T^2}$$

The constant $4\pi^2$ is exact (no uncertainty), so it does not contribute.

Approach

1. Write the proportionality $g \propto l/T^2$.
2. Add the percentage uncertainty from $l$.
3. Add twice the percentage uncertainty from $T$ because of the square.

Step-by-Step Reasoning

From

$$g = \frac{4\pi^2 l}{T^2}$$

ignore the constant and focus on variables:

$$g \propto \frac{l}{T^2}$$

Given:

• $l$ has $\pm 2\%$
• $T$ has $\pm 3\%$

Because $T$ is squared, its percentage uncertainty contribution doubles:

$$\%\Delta(T^2) = 2(\%\Delta T) = 2(3\%) = 6\%$$

Now add contributions for division:

$$\%\Delta g = 2\% + 6\% = 8\%$$

Key Takeaways

• For $y = a^m b^n / c^p$, percentage uncertainty is $m\%\Delta a + n\%\Delta b + p\%\Delta c$.
• Constants do not contribute to measurement uncertainty.

Common Mistakes

• Subtracting percentage uncertainties for division (they should add).
• Forgetting to multiply the uncertainty in $T$ by $2$ because of $T^2$.
• Including uncertainty in $4\pi^2$.

Things to Be Careful About

• These rules assume uncertainties are small and independent, which is the standard A-Level approach.
• Use the power on the variable exactly as it appears in the equation (here $T^2$ not $T$).

Background Concept

Percentage uncertainty tells you the uncertainty relative to the measured (or calculated) value. To find the absolute uncertainty:

$$\Delta x = \left(\frac{\%\Delta x}{100}\right) x$$

Absolute uncertainty has the same unit as the quantity itself.

Understanding the Question

You have already found $g$ in (b)(i) and the percentage uncertainty in $g$ in (b)(ii). You must combine them to get the absolute uncertainty in $g$.

Approach

1. Convert the percentage uncertainty into a decimal (fractional) uncertainty.
2. Multiply by the calculated value of $g$.
3. State the result with unit $\text{m s}^{-2}$.

Step-by-Step Reasoning

From earlier parts:

• $g = 9.63\ \text{m s}^{-2}$
• percentage uncertainty in $g$ is $8\%$

Convert $8\%$ to a fraction:

$$8\% = \frac{8}{100} = 0.08$$

Now multiply to get absolute uncertainty:

$$\Delta g = 0.08 \times 9.63 = 0.7704\ \text{m s}^{-2}$$

Round appropriately:

$$\Delta g \approx 0.77\ \text{m s}^{-2}$$

Key Takeaways

• Absolute uncertainty = (percentage uncertainty / 100) (\times) value.
• Absolute uncertainty carries the same units as the quantity.

Common Mistakes

• Writing the absolute uncertainty as $8\ \text{m s}^{-2}$ (forgetting to divide by $100$).
• Giving the uncertainty without units.
• Using $g = 9.8$ instead of the calculated value from (b)(i) when the question says to use your answers.

Things to Be Careful About

• Keep at least 2 significant figures in the uncertainty unless the mark scheme specifies otherwise.
• Ensure you use the same $g$ value you calculated in (b)(i) (error-carried-forward is usually allowed, but consistency is key).