How Variables Scale in Equations

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You will have used equations all the time in science and maths, usually by plugging in numbers. For example, if we look at Newton’s second law

\(F = m a\)

and you know that mass is 1 kilogram and acceleration is 2 \(ms^{-2}\) then the force is \(1 \times 2 = 2\) newtons. But there’s something else you can do with equations that is incredibly useful and often crops up in exams, particularly in GCSE Mathematics and A-level Physics. This is spotting how variables scale.

What does scaling mean? Well, instead of putting in actual numbers into an equation you consider what would happen if the variables on the right-hand side of the equals sign are scaled up or down by a factor. Using Newton’s equation we can see that if you double the mass on the right-hand side you would double the force. Or if you double the acceleration you would double the force. This kind of scaling is the simplest type: it’s linear. The remaining examples in this post are more interesting because they aren’t linear.

Why Godzilla Couldn’t Walk

The exam question that occurs all the time is scaling of area and volume with length. Let’s say we have a scaling gun that scales up animals. To create Godzilla we will breed a mini-Godzilla and then use the scaling gun to scale him up. Let’s think what will happen to four things:

  • Godzilla’s height
  • Godzilla’s surface area
  • Godzilla’s volume
  • Godzilla’s mass

Mini-Godzilla is a Tyrannosaurus rex, 10 metres high because we cloned him along the lines of Jurassic Park. Then we scale his height by a factor of ten. He is now 100 metres high!

Area Scales With Length Squared

Now the surface area of a solid depends on the square of its dimensions. For example, a regular cube has six sides and if each edge has length \(l\) then the surface area

\(A=6 l^2\)

Now if we increase \(l\) by a factor of ten the surface area \(A\) will increase by a factor of 100, which is 10 squared. The same is true of a sphere. If its radius is \(r\) then its surface area is

\(A = 4 \pi r^2\)

which, if we increase the radius by a factor of ten, will increase by a factor of 100, which is 10 squared.

It’s true of a cube, a sphere, or Godzilla. If we increase an object’s dimensions by a factor of 10 its area will increase by a factor of 100.

Volume Scales With Length Cubed

What about Godzilla’s volume? If we think about our cube again its volume is

\(V=l^3\)

which increases by a factor of 1000 if we increase the cube’s dimensions by ten. The same is true of a sphere with volume

\(V=\frac{4}{3} \pi r^3\)

which, if we increase the radius by a factor of 10, has its volume increase by a factor of 1000. So Godzilla’s volume will also increase by a factor of 1000.

Mass Scales With Length Cubed

If the scaling gun doesn’t change the density of Godzilla then his mass will also scale up by a factor of 1000. This is a problem. Think of the stress on Godzilla’s legs. Stress is

\(\mbox{stress}=\frac{\mbox{force}}{\mbox{area}}\)

The force is Godzilla’s weight and the area is the cross-sectional area of Godzilla’s legs. Weight goes up by a factor of 1000, the area goes up by a factor of 100 so the stress on Godzilla’s legs will increase by a factor of 10. His bones would shatter under his own weight!

If we think about his ability to radiate heat this would also be problematic. This is because he radiates heat through his surface, which has gone up by a factor of 100 but heat is generated throughout his body and that has scaled by a factor of 1000, so his surface area to volume ratio has decreased by a factor of 10. Godzilla’s legs would shatter and he would always be overheating. Maybe we shouldn’t use that scaling gun after all.

Scaling Units

Exam questions often require you to scale from centimetres to metres but many people get confused when scaling areas from centimetres squared to metres squared or centimetres cubed to meters cubed. This works in exactly the same way as the Godzilla example above:

  • Area:
    • 1 m = 100 cm
    • \(1 \mbox{m}^2 = 100^2\mbox{cm}^2 = 10,000 \mbox{cm}^2\)
    • 1 m = 1000 mm
    • \(1 \mbox{m}^2 = 1000^2 \mbox{mm}^2 =1,000,000 \mbox{mm}^2\)
  • Volume
    • 1 m = 100 cm
    • \(1 \mbox{m}^3 = 100^3 \mbox{cm}^3 =1,000,000 \mbox{cm}^3\)
    • 1 m = 1000 mm
    • \(1 \mbox{m}^3 = 1000^3 \mbox{mm}^3 = 1,000,000,000 \mbox{mm}^3\)

To go from the length scaling factor to the area scaling factor you square the ratio.

To go from the length scaling factor to the volume scaling factor you cube the ratio.

To go from the area scaling factor to the volume scaling factor you first take the square root of the ratio (to turn it into a length ratio) then cube the ratio.

To go from the volume scaling factor to the area scaling factor you first take the cube root of the ratio (to turn it into a length ratio) then square the ratio.

Inverse Scaling

Sometimes quantities scale down as another quantity scales up, a relationship called inverse scaling.

Travel time scales as the inverse of the speed

\(\mbox{Time}=\frac{\mbox{Distance}}{\mbox{Speed}}\)

Density scales as the inverse of the volume

\(\mbox{Density}=\frac{\mbox{Mass}}{\mbox{Volume}}\)

Gravitational potential energy scales as the inverse of the distance from a mass \(M\) with gravitational constant \(G\) is

\(\mbox{Gravitational Potential Energy}=-\frac{GM}{r}\)

The power of a lens in diopters scales as the inverse of the focal length \(f\) of the lens

\(\mbox{P}=\frac{1}{f}\)

Quadratic Scaling

There are lots of examples of equations where the scaling isn’t linear, it goes as the square of the scaling factor. Here are some examples:

Kinetic energy scales as the square of the velocity

\(\mbox{Kinetic Energy}=\frac{1}{2} m v^2\)

Energy stored in a spring with spring constant \(k\) scales as the square of the extension \(x\) \(\mbox{Spring Energy} = \frac{1}{2} k x^2 \)

Maximum acceleration for a simple harmonic oscillator with amplitude \(A\) scales as the square of angular velocity \(\omega\) \(\mbox{Maximum Acceleration} = A \omega^2 \)

Power dissipated in a resistor of resistance \(R\) scales as the square of current \(I\) or voltage \(V\) \(\mbox{Power} = I^2 R = \frac{V^2}{R} \)

Square Root Scaling

These seem trickier but instead of squaring you take the square root of the scaling factor.

The period of a pendulum with gravitational acceleration \(g\) scales as the square root of the pendulum length \(l\) \(T \approx 2\pi \sqrt{\frac{l}{g}}\)

The period of a mass on a spring \(g\) scales as the square root of the mass \(m\) \(T = \frac{1}{2\pi} \sqrt{\frac{m}{k}}\)

Escape velocity from a body with radius \(r\) scales as the square root of the mass of the planet \(M\) \(v_{\mbox{escape}} = \sqrt{\frac{2GM}{r}}\)

The root mean square velocity of gas molecules of mass \(m\) in an ideal gas  scales as the square root of the temperature \(T\) \(v_{\mbox{rms}} = \sqrt{\frac{3kT}{m}}\)

Worksheet

Here are 11 questions that you can use to test your ability to find how equations scale:

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About the author

ramin

Ramin is a science and maths tutor. He has a first class degree and a doctorate in Physics. He has written two finance books and also teaches adults about investment through his company PensionCraft.