The Pizza and Proportional Reasoning

J. A. Hester

113 The Pizza and Proportional Reasoning

J. A. Hester

The Pizza

Buy the bigger Pizza!

Proportional Reasoning

For this class, we’ll be working with proportional or multiplicative reasoning with formulas that are not directly proportional.

Examples of directly proportional relationships are the weight of apples purchased and the cost of the apples or the time driving a constant speed and the miles driven. In these relationships, doubling the independent variable (weight of apples or time driven) also doubles the dependent variable (cost of apples or distance driven).

We will also work with relationships in which the dependent variable is proportional to the square of the independent variable (e.g., surface area), the inverse of the independent variable (e.g., Wien’s law), or even the inverse of the square of the independent variable (e.g. the brightness-distance relationship).

There will be proportional reasoning used throughout the course, so it’s worth taking the time to learn this skill. To help, here are some examples.

Wien’s law

(1) $\begin{equation*} \lambda_{peak}=\frac{2900\mu \rm{K}}{\rm{T}} \end{equation*}$

The peak wavelength of a thermal spectrum ( $\lambda_{peak}$ ) is inversely proportional to the temperature of the object emitting the spectrum ( $\rm{T}$ ).

If the temperature of the source doubles (times 2), the inverse of the temperature is halved (times 1/2). Since the peak wavelength is proportional to the inverse of the temperature, it will be halved.

If the temperature of the source decreases to a third of its original value (times 1/3), the inverse of the temperature will triple (the inverse of 1/3 is 3/1, or just 3). The peak wavelength will also triple. Note that this makes some sense as cooler objects are redder (longer wavelengths).

Stefan-Boltzmann Law

(2) $\begin{equation*} f=\sigma \rm{T}^4 \end{equation*}$

The flux, $f$ , from an object’s surface is proportional to its temperature to the fourth power, $\rm{T}^4$ .

If the temperature of the object doubles (times 2), its temperature to the fourth power increases by a factor of 16 (2⁴=16). The flux (the energy radiated by each square meter of the object’s surface) will therefore also increase by a factor of 16. So if the original flux were 6 Watts per square meter (W/m²), then the new flux would be 16 times this (96 W/m²) even though the temperature only doubled.

If the temperature of the object is halved (times 1/2), the flux will be decreased to 1/16 of it’s original value.

Brightness and Distance

(3) $\begin{equation*} b=\frac{1}{4\pi r^2}L \end{equation*}$

The brightness of a source of light is proportional to the inverse of the square of the distance between the observer and the source. These “inverse square laws” are common in physics and astrophysics. In the formula above, $b$ is the source’s brightness, $L$ is its luminosity, and $r$ is the distance to the source.

If the distance to a source is doubled (times 2), then the square of the distance will be quadrupled (2²=4, this is the square part) and the inverse of the square will be a fourth of its original value (1/4, this is the inverse part). Therefore, the source will appear to be a quarter as bright after it is moved twice as far away.

If the distance to the source were quartered (1/4 the original value), the square of the distance would decrease to 1/16 of its original value (1/4²) and the inverse of the square would increase by a factor of 16 (the inverse of 1/16 is 16). The light would therefore be 16 times as bright.

This last example makes sense because the closer an object is, the brighter it appears, but please note that the relationship is not linear. Bringing an object four times closer does not make it four times brighter, it makes it 16 times brighter.

Surface Gravity

(4) $\begin{equation*} g=G\frac{M}{r^2} \end{equation*}$

The surface gravity of a planet, moon, star, or any spherical body is (directly) proportional to the mass of the body and inversely proportional to the square of the radius of the body. (Another inverse square law!) In the formula above, $g$ is the surface gravity, $G$ is the Universal Gravitational Constant (a number, with appropriate units, that is the same everywhere in the Universe), $M$ is the mass of the planet, and $r$ is the radius of the planet.

If the Earth were twice as massive, but its radius didn’t change (the Earth would be denser in this case), then its surface gravity would be twice as high. If the Earth’s mass were unchanged, but its radius were doubled (the density would decrease), then the surface gravity would be only a quarter of what it is. The square of 2 (doubled) is 4, and the inverse of 4 is 1/4.

You may have heard that your weight on the Moon is about a sixth (or 16%) of what it is on the Earth. The Moon’s mass is about 1.2% of the Earths. So if the Earth and the Moon were the same size, you’d expect that the Moon’s surface gravity would be only 1.2% of the Earth’s. But the Moon isn’t the same size as the Earth, it’s smaller. The Moon’s radius is about 27% of the Earth’s. If the Moon were as massive as the Earth, it’s surface gravity would be (1/0.27)^2, or about 13.7 times stronger than Earth’s.

To find the real comparison between the Earth and the Moon, you need to multiply these two effects together. The Moon’s gravity is (0.012)(13.7), or about 16%, of the Earth’s. So a 100 lb object on the Earth would only weight 16 lb on the Moon.