Ratios and proportional reasoning Knowledge-Based Study

Study notes

1. What is a Ratio?

   In physics, a ratio is a quantitative comparison of two values or physical quantities. It indicates how many times one value contains or is contained within another.

   Ratios can compare quantities with identical dimensions or entirely different dimensions:

   • Dimensionless ratios: Comparing quantities of the same dimension, such as efficiency (useful work divided by total work) or the coefficient of friction. In these cases, the units cancel out completely.
   • Rates and derived quantities: Comparing quantities with different units, such as distance per unit time (speed, measured in m/s) or mass per unit volume (density, measured in kg/m3).

   Expressing relationships as ratios allows physicists to compare systems regardless of absolute scale.

   Forms of Proportionality

   Proportional reasoning analyzes how a change in one physical quantity causes a predictable change in another when all other variables remain constant.

   1. Direct Proportionality

   Two quantities $y$ and $x$ are directly proportional ($y \propto x$) if their ratio remains constant:

   $$\frac{y}{x} = k \implies y = kx$$

   If $x$ is multiplied by a factor of $n$, then $y$ is also multiplied by $n$. For example, under constant acceleration $a$, velocity gained from rest is directly proportional to time ($v = at$). Doubling the time doubles the final velocity.

   2. Inverse Proportionality

   Two quantities $y$ and $x$ are inversely proportional ($y \propto \frac{1}{x}$) if their product remains constant:

   $$y \cdot x = k \implies y = \frac{k}{x}$$

   If $x$ is multiplied by a factor of $n$, $y$ is divided by $n$. For instance, for an ideal gas at constant temperature, pressure is inversely proportional to volume ($P = \frac{k}{V}$). Halving the volume doubles the pressure.

   Nonlinear Proportional Relationships

   Many foundational physical laws involve powers or roots rather than simple linear dependencies.

   Square and Inverse-Square Dependencies

   • Direct Square: The kinetic energy of an object is proportional to the square of its speed, $K \propto v^2$. If the speed triples (factor of 3), the kinetic energy increases by a factor of $3^2 = 9$.
   • Inverse-Square Law: Gravitational and electrostatic forces decrease with the square of the distance between centers, $F \propto \frac{1}{r^2}$. If the distance between two charges is doubled, the electric force decreases to $\frac{1}{2^2} = \frac{1}{4}$ of its initial value.

   Recognizing the mathematical exponent of each variable enables rapid qualitative and quantitative predictions without recalculating constants.

   The Ratio Method for Multi-Variable Equations

   When multiple variables change simultaneously, writing the ratio of the final state to the initial state simplifies calculation and cancels constant factors.

   General Strategy

   1. Identify the governing physical equation and state which parameters are held constant.
   2. Write the formula for both the initial state (state 1) and the final state (state 2).
   3. Divide the second equation by the first equation to eliminate invariant constants.
   4. Substitute the known scaling factors to determine the unknown ratio.

   Worked Example

   The period of a simple pendulum is given by $T = 2\pi\sqrt{\frac{L}{g}}$. How does the period change if the length $L$ is increased by a factor of 4 while local gravity $g$ remains constant?

   $$\frac{T_2}{T_1} = \frac{2\pi\sqrt{\frac{L_2}{g}}}{2\pi\sqrt{\frac{L_1}{g}}} = \sqrt{\frac{L_2}{L_1}} = \sqrt{4} = 2$$

   The period doubles. Using proportional reasoning avoids the need to know the exact numerical values of length or gravitational acceleration.