Difficulty: Introductory | Prerequisites: Basic algebra, logarithm rules
These topics sit right at the start of PHYSICS 114. Scaling and estimation teach you to think about how physical quantities change when you change a system's size, which underpins nearly every calculation in the course. Log-log graphs show up when data follows a power law, and reading them correctly is a skill tested early and reused throughout. If you are comfortable with exponents and logarithms, you are ready.
Physical quantities like area scale nonlinearly with length: doubling a pizza's diameter quadruples its area. When data follows a power law, plotting log(y) vs log(x) turns the curve into a straight line whose slope gives the exponent. You can read ratios directly off these graphs without converting back to raw numbers, which saves time on the exam.
Scaling (dimensional analysis)
The study of how a physical quantity changes when you change the size of a system. In simple terms, this means asking: if I make something twice as big, does the quantity double, quadruple, or change by some other factor?
Area scaling
Area is proportional to the square of a linear dimension. A circle's area goes as the diameter squared, so a 16-inch pizza has (16/10)^2 = 2.56 times the area of a 10-inch pizza. Think of it as: length doubles, area quadruples.
Log-log plot
A graph where both axes show the logarithm (base 10) of the variable rather than the variable itself. When data follows a power law (y = ax^n), a log-log plot turns the curve into a straight line with slope equal to the exponent n. In simple terms, it is a way of making curved power-law data look linear so you can read the relationship at a glance.
Power law
A relationship of the form y = ax^n, where n is the exponent. On a log-log plot this becomes log(y) = n log(x) + log(a), a straight line with slope n and intercept log(a).
Specific metabolism
Metabolic rate divided by body mass (watts per kilogram). Smaller animals have higher specific metabolism than larger ones, and the relationship follows a power law when plotted against mass.
Area of a circle scales with the square of the diameter: A = pi(d/2)^2, so A is proportional to d^2.
To compare two pizzas, you do not need the full area. The ratio of areas equals the ratio of diameters squared: A1/A2 = (d1/d2)^2.
For the pizza problem: one 10-inch pizza feeds 2 people. That is 50 square inches of pizza per person (area = pi(5)^2 = 78.5 sq in, divided by 2). Ten guests need 10 x 39.3 = 393 sq in. A 16-inch pizza has area pi(8)^2 = 201 sq in. So you need 393/201 = 1.95, which rounds up to 2. But check: each 16-inch pizza has (16/10)^2 = 2.56 times the area of a 10-inch, so it feeds 2 x 2.56 = 5.12 people. Ten guests need 10/5.12 = 1.95, so 2 pizzas. Answer: B) 2.
A log-log plot has log(x) on the horizontal axis and log(y) on the vertical axis.
A straight line on a log-log plot means the underlying relationship is a power law: y = a * x^n, where n is the slope of the line.
The y-intercept of the line gives log(a).
The regression line in the exam graph is: log(specific metabolism) = -0.32566 * log(mass in kg) + 0.24808.
To find the ratio of specific metabolism for two animals, plug in each animal's mass (in kg), compute log(SM) for each, then take the difference.
Mouse: mass = 30 g = 0.030 kg, so log(0.030) = -1.5229. Plug in: log(SM_mouse) = -0.32566(-1.5229) + 0.24808 = 0.4441 + 0.2481 = 0.7441.
Elephant: mass = 2.2 x 10^4 kg, so log(2.2 x 10^4) = 4.3424. Plug in: log(SM_elephant) = -0.32566(4.3424) + 0.24808 = -1.1663.
The difference: log(SM_mouse) - log(SM_elephant) = 0.7441 - (-1.1663) = 1.9104.
This difference equals log(ratio), so ratio = 10^1.9104 = approximately 81. Answer: A) 81.
log(y1) - log(y2) = log(y1/y2). So subtracting two y-values on a log-log plot gives you the log of their ratio directly. Raise 10 to that power to get the ratio.
A = \pi \left(\frac{d}{2}\right)^2 = \frac{\pi d^2}{4}\frac{A_1}{A_2} = \left(\frac{d_1}{d_2}\right)^2\log(y) = n \cdot \log(x) + \log(a) \quad \text{(power law on a log-log plot)}\log\left(\frac{y_1}{y_2}\right) = \log(y_1) - \log(y_2)Scaling analysis is how engineers decide whether a lab prototype will behave the same way at full size. If you are designing a heat sink or a drug delivery particle, area-to-volume ratios determine performance. Log-log plots are used routinely in biology (metabolic scaling), astronomy (luminosity vs mass), and materials science (crack propagation rates).
Students often think that doubling the diameter of a pizza doubles the amount of food. It does not. It roughly quadruples the area, so a 16-inch pizza is not 1.6 times a 10-inch pizza; it is 2.56 times.
Students sometimes try to convert log-log graph values back to raw numbers before computing a ratio. You do not need to. Subtract the two log values and raise 10 to that power.
Students confuse the slope of the log-log line with the ratio itself. The slope gives the exponent of the power law, not the ratio of any two data points.
On the exam, students forget to convert grams to kilograms before taking the log. The graph axis is log(mass in kg), so 30 g must become 0.030 kg.
Scaling and estimation appear on nearly every PHYSICS 114 exam. Expect at least one question asking you to compare areas or volumes of differently sized objects.
Log-log graph questions almost always ask for a ratio. Practise the subtract-and-raise method until it is automatic.
The exam provides the regression equation on the graph. Use it directly rather than trying to read individual points off the line.
True or false: A 20-inch pizza has twice the area of a 10-inch pizza. (False. It has four times the area.)
On a log-log plot, a straight line with slope -0.33 means y is proportional to x raised to what power? (-0.33)
Fill in the blank: log(y1) - log(y2) = log(______). (y1/y2)
True or false: To find a ratio from a log-log graph, you must first convert the log values back to raw numbers, then divide. (False. Subtract the log values, then raise 10 to that difference.)
If log(mass in kg) = -1.5, what is the mass in grams? (10^-1.5 kg = 0.0316 kg = 31.6 g)
Q: A 10-inch pizza feeds 2 people. How many 16-inch pizzas do you need for 10 guests?
A: A 16-inch pizza has (16/10)^2 = 2.56 times the area of a 10-inch pizza, so it feeds 2 x 2.56 = 5.12 people. For 10 guests: 10/5.12 = 1.95, rounded up to 2. Answer: 2.
Q: Using the regression line log(SM) = -0.32566 log(m) + 0.24808, find the ratio of specific metabolism for a 30 g mouse to a 2.2 x 10^4 kg elephant.
A: Mouse mass in kg = 0.030, log(0.030) = -1.523. Elephant log(2.2 x 10^4) = 4.342. Difference in log(SM) = [-0.32566(-1.523) + 0.24808] - [-0.32566(4.342) + 0.24808] = -0.32566(-1.523 - 4.342) = -0.32566(-5.865) = 1.910. Ratio = 10^1.91 = 81.
Q: A log-log plot of metabolic rate vs body mass has a best-fit line with slope 0.75. What does this tell you about how metabolic rate scales with mass?
A: Metabolic rate is proportional to mass raised to the 0.75 power. Doubling an animal's mass increases its metabolic rate by a factor of 2^0.75 = 1.68, not 2.
Scaling connects directly to unit analysis and dimensional analysis, which you will use throughout every kinematics and dynamics problem. The habit of checking whether an answer "scales right" (e.g. doubling speed should quadruple kinetic energy) is one of the most useful error-catching tools in physics.
Log-log plots reappear when studying sound intensity (decibels), earthquake magnitudes (Richter scale), and any power-law distribution in data analysis.
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