12 Motion in a Circle

Here is the note written using standard CIE 9702 Physics and 9709 Maths notation and terminology.

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Graph Scale Derivation (CIE 9702 / 9709)

Definitions

• $N$ = total number of large squares on the axis (usually $8$ or $12$).
• $R$ = range of data to be plotted ($R=x_{\text{max}}-x_{\text{min}}$).
• $s$ = value represented by $1$ large square.

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The Two Graph Rules (9702 Mark Scheme)

1. Points must fit on the grid: $\text{Total axis range}\ge \text{Data range}$ $N\times s\ge R⟹s\ge \frac{R}{N}$

2. Points must occupy at least half the grid (50% rule): $\text{Squares used}\ge \frac{N}{2}$ $\frac{R}{s}\ge \frac{N}{2}⟹s\le \frac{2R}{N}$

Combining both gives the acceptable range for the scale $s$: $\frac{R}{N}\le s\le \frac{2R}{N}$

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Standard Scales Rule

The mark scheme requires sensible scales: $s\text{ must be }1\times 10^n,2\times 10^n,\text{or}5\times 10^n$ (Awkward scales like $3,7,6,$ or $2.5$ lose marks).

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Derivation of the “Danger Zone”

We want to find when no standard scale $s$ can satisfy $\frac{R}{N}\le s\le \frac{2R}{N}$.

Notice the ratio between consecutive standard scales:

• From $1$ to $2$: factor of $\frac{2}{1}=2.0$
• From $5$ to $10$: factor of $\frac{10}{5}=2.0$
• From $2$ to $5$: factor of $\frac{5}{2}=2.5$

Because our acceptable range only allows a factor of $2$ ($\text{upper limit}=2\times \text{lower limit}$), a problem occurs only between $2\times 10^n$ and $5\times 10^n$.

For the entire interval to fall in the gap between $2\times 10^n$ and $5\times 10^n$:

1. The lower limit must be greater than $2\times 10^n$: $\frac{R}{N}>2\times 10^n$

2. The upper limit must be less than $5\times 10^n$: $\frac{2R}{N}<5\times 10^n⟹\frac{R}{N}<2.5\times 10^n$

Combining these two inequalities: $2\times 10^n<\frac{R}{N}<2.5\times 10^n$

Multiply everything by $N$: $2N\times 10^{\mathbf{n}}<\mathbf{R}<2.5N\times 10^{\mathbf{n}}$

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Summary for Exam Planning (9702 P3 / P5)

If your range $R$ falls in this interval, you cannot choose a standard scale that obeys both marking points.

• For $N=8$ squares: $16\times 10^n<R<20\times 10^n$ (Avoid ranges whose leading digits are between $1.6$ and $2.0$)

• For $N=12$ squares: $24\times 10^n<R<30\times 10^n$ (Avoid ranges whose leading digits are between $2.4$ and $3.0$)