Stating a theoretical relationship and predicting the outcome

Key ideas

A physics theory, for this standard, is an established relationship (often derived from physics you already know) predicting how the dependent variable should depend on the independent variable — for example, theory predicts a pendulum's period $T$ relates to its length $L$ by

$T = 2\pi\sqrt{\frac{L}{g}}$

Before collecting data, restate the theory as a power relationship $y = kx^n$ and identify the value of $n$ the theory predicts. For the pendulum, $T \propto \sqrt{L} = L^{0.5}$ , so the theory predicts $n = 0.5$ .
This predicted $n$ becomes your benchmark — the whole investigation exists to test whether the experimental gradient of a $\log y$ vs $\log x$ graph is consistent with this predicted value.
State clearly, in your plan, what result would support the theory and what result would not — e.g. "if the theory holds, the log–log gradient should be close to $0.5$ ; a gradient clearly outside that range (accounting for uncertainty) would not support it."

Identifying the theoretical constant

The theory usually also predicts what the constant $k$ (the y-intercept of a log-log plot, "undone" from log form) should physically represent.
For the pendulum, $T = 2\pi\sqrt{\dfrac{1}{g}}\cdot\sqrt{L}$ , so the constant multiplying $\sqrt{L}$ is $k = 2\pi\sqrt{1/g}$ — meaning the experimental value of $k$ can be used to calculate an experimental value of $g$ , which is itself a second way to test the theory (compare your $g$ to the accepted $9.81$ m s⁻²).

Worked ExamplePredicting the log-log gradient for a pendulum

Theory predicts $T = 2\pi\sqrt{L/g}$ . State the power $n$ this predicts for a log-log plot of $T$ against $L$ , and explain how the constant $k$ found from the graph could be used to test the theory further.

Step 1 — Rewrite the theory as a power law

$T = 2\pi\sqrt{\frac{L}{g}} = \left(2\pi\sqrt{\frac{1}{g}}\right) L^{0.5}$

This has the form $T = kL^n$ with $n = 0.5$ and $k = 2\pi\sqrt{1/g}$ .

Step 2 — State the predicted gradient

On a $\log T$ vs $\log L$ plot, the gradient equals $n$ , so the theory predicts a gradient of $\mathbf{0.5}$ .

Step 3 — Explain the further test using $k$

The experimental $k$ (found from the graph's y-intercept, $k = 10^{c}$ ) can be rearranged, since $k = 2\pi\sqrt{1/g}$ :

$g = \frac{4\pi^2}{k^2}$

Calculating $g$ this way and comparing it to the accepted value $9.81$ m s⁻² gives a second, independent check of the theory — the power confirms the shape of the relationship, and the constant confirms the physics behind it.

Practice question

Theory predicts that the range $R$ of a projectile launched at a fixed speed depends on launch angle $\theta$ as $R \propto \sin(2\theta)$ . Explain why this particular theory is not well suited to the log-log linearising method used elsewhere on this page.

Worked solution: The log-log method (plotting $\log y$ vs $\log x$ ) only linearises power relationships of the form $y = kx^n$ . $R \propto \sin(2\theta)$ is not a power relationship — $\theta$ appears inside a trigonometric function, not raised to a power — so taking logs of $R$ and $\theta$ would not produce a straight line. A different analysis (e.g. plotting $R$ directly against $\sin(2\theta)$ , which should be linear through the origin) would be needed instead.

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

Theory predicts that the energy stored in a spring $E$ depends on its extension $x$ as $E = \tfrac{1}{2}kx^2$ . State the power $n$ this predicts for a log-log plot of $E$ against $x$ .

Merit

Explain why stating the predicted power $n$ before collecting data is better practice than only working it out afterwards.

Excellence

An investigation into $T = 2\pi\sqrt{L/g}$ finds a log-log gradient of $0.52 \pm 0.04$ and an experimental $g$ of $9.6 \pm 0.5$ m s⁻². Evaluate whether this data supports the theory.

Stating a theoretical relationship and predicting the outcome

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Key ideas

A physics theory, for this standard, is an established relationship (often derived from physics you already know) predicting how the dependent variable should depend on the independent variable — for example, theory predicts a pendulum's period $T$ relates to its length $L$ by

$T = 2\pi\sqrt{\frac{L}{g}}$

Before collecting data, restate the theory as a power relationship $y = kx^n$ and identify the value of $n$ the theory predicts. For the pendulum, $T \propto \sqrt{L} = L^{0.5}$ , so the theory predicts $n = 0.5$ .
This predicted $n$ becomes your benchmark — the whole investigation exists to test whether the experimental gradient of a $\log y$ vs $\log x$ graph is consistent with this predicted value.
State clearly, in your plan, what result would support the theory and what result would not — e.g. "if the theory holds, the log–log gradient should be close to $0.5$ ; a gradient clearly outside that range (accounting for uncertainty) would not support it."

Identifying the theoretical constant

The theory usually also predicts what the constant $k$ (the y-intercept of a log-log plot, "undone" from log form) should physically represent.
For the pendulum, $T = 2\pi\sqrt{\dfrac{1}{g}}\cdot\sqrt{L}$ , so the constant multiplying $\sqrt{L}$ is $k = 2\pi\sqrt{1/g}$ — meaning the experimental value of $k$ can be used to calculate an experimental value of $g$ , which is itself a second way to test the theory (compare your $g$ to the accepted $9.81$ m s⁻²).

Worked ExamplePredicting the log-log gradient for a pendulum

Step 1 — Rewrite the theory as a power law

$T = 2\pi\sqrt{\frac{L}{g}} = \left(2\pi\sqrt{\frac{1}{g}}\right) L^{0.5}$

This has the form $T = kL^n$ with $n = 0.5$ and $k = 2\pi\sqrt{1/g}$ .

Step 2 — State the predicted gradient

On a $\log T$ vs $\log L$ plot, the gradient equals $n$ , so the theory predicts a gradient of $\mathbf{0.5}$ .

Step 3 — Explain the further test using $k$

The experimental $k$ (found from the graph's y-intercept, $k = 10^{c}$ ) can be rearranged, since $k = 2\pi\sqrt{1/g}$ :

$g = \frac{4\pi^2}{k^2}$

Practice question

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

Theory predicts that the energy stored in a spring $E$ depends on its extension $x$ as $E = \tfrac{1}{2}kx^2$ . State the power $n$ this predicts for a log-log plot of $E$ against $x$ .

Merit

Explain why stating the predicted power $n$ before collecting data is better practice than only working it out afterwards.

Excellence

An investigation into $T = 2\pi\sqrt{L/g}$ finds a log-log gradient of $0.52 \pm 0.04$ and an experimental $g$ of $9.6 \pm 0.5$ m s⁻². Evaluate whether this data supports the theory.

Stating a theoretical relationship and predicting the outcome

Key ideas

Identifying the theoretical constant

Step 1 — Rewrite the theory as a power law

Step 2 — State the predicted gradient

Step 3 — Explain the further test using kkk

Practice question

Test yourself

Practice by grade

Stating a theoretical relationship and predicting the outcome

Key ideas

Identifying the theoretical constant

Step 1 — Rewrite the theory as a power law

Step 2 — State the predicted gradient

Step 3 — Explain the further test using kkk

Practice question

Test yourself

Practice by grade

Step 3 — Explain the further test using $k$

Step 3 — Explain the further test using $k$