Mars goes round the Sun at \(1.52\) times the Earth's distance. How long is its year? The chapter Gravitational Energy and Orbits gives the period as \(T^2 = (4\pi^2/GM)\,a^3\), where \(M\) is the Sun's mass. The question seems to need \(G\) and \(M\). It needs neither, because the same constant belongs to every planet of the Sun.
Proportional means a fixed ratio
Two quantities are proportional when their ratio stays the same. The blades of a ceiling fan all turn at one angular speed \(\omega\), so the speed of a point is \(v = r\omega\). A point twice as far from the axis moves twice as fast, because \(v/r = \omega\) is the same for every point. We write \(v \propto r\), and the fixed ratio \(\omega\) is the constant of proportionality.
A quantity \(y\) is proportional to the \(n\)th power of \(x\), written \(y \propto x^n\), when \(y = kx^n\) with the same constant \(k\) in every case compared. The power \(n\) may be any real number: negative for an inverse law, a fraction for a root.
\[ y_1 = kx_1^n, \qquad y_2 = kx_2^n \]
Why. the same law, with the same constant \(k\), holds in both cases.
\[ \dfrac{y_2}{y_1} = \dfrac{k\,x_2^n}{k\,x_1^n} \]
Why. divide the second equation by the first.
\[ \dfrac{y_2}{y_1} = \left(\dfrac{x_2}{x_1}\right)^n \]
Read it. \(k\) cancels, and a ratio of two \(n\)th powers is the \(n\)th power of the ratio.
The constant has cancelled, and so has every unit. So a ratio question never needs the constant: not \(G\), not \(M\), not the viscosity of air. The words "if \(x\) doubles" mean \(x_2/x_1 = 2\), and then \(y\) changes by the factor \(2^n\). If \(x\) is multiplied by any factor \(k\), \(y\) is multiplied by \(k^n\).
The gravitational force between two bodies varies as \(1/r^2\). The distance between them is tripled. What happens to the force?
Direct, inverse and inverse square
A handful of powers covers most of physics. In a direct law, \(y \propto x\), both grow together in the same ratio. In an inverse law, \(y \propto 1/x\), one falls as the other grows. At a fixed temperature, a gas squeezed to a third of its volume has three times the pressure. In an inverse-square law, \(y \propto 1/x^2\), doubling the distance cuts the effect to a quarter. Gravity, the force between charges and the intensity of a lamp all fall off this way.
The terminal velocity of a small raindrop varies as the square of its radius. A drop of twice the radius falls at how many times the speed?
Higher powers magnify a change
The higher the power of a power law, the more a change is magnified. In the figure below, a slider multiplies a radius \(r\) by a factor \(k\). At \(k = 2\) the fourth power grows sixteen times. At \(k = 0.8\), a cut of only 20 %, the fourth power falls to \(0.41\) of its old value. This is why a narrowed artery or a pinched drip tube changes the flow so much: the flow through a tube grows as \(r^4\).
Several factors multiply
A law may depend on several quantities. Then each quantity brings its own factor, and the factors multiply. The period of a pendulum is \(T = 2\pi\sqrt{l/g}\), so \(T \propto l^{1/2}g^{-1/2}\). Take a pendulum four times as long to a place where \(g\) is half as large. Its period changes by \(4^{1/2} \times (1/2)^{-1/2} = 2 \times 1.41 = 2.83\).
Back to the planets
Kepler's third law says \(T^2 \propto a^3\) for every planet of the Sun, so \(T \propto a^{3/2}\). Measure distances in units of the Earth's orbit, astronomical units, and periods in Earth years. Then the Earth has \(a = 1\) and \(T = 1\), and every other planet has \(T = a^{3/2}\) exactly. The figure below shows what this means: a planet twice as far away takes \(2^{3/2} = 2.83\) years, so the Earth laps it almost three times.
Mars orbits the Sun at \(a = 1.52\,\text{AU}\) and Jupiter at \(5.2\,\text{AU}\). Find the length of the year of each planet, in Earth years.
Kepler's third law is \(T^2 = (4\pi^2/GM)\,a^3\). The constant \(4\pi^2/GM\) depends only on the Sun, so it is the same for all three planets.
Compare each planet with the Earth: \(T/T_E = (a/a_E)^{3/2}\), with \(T_E = 1\) year and \(a_E = 1\,\text{AU}\).
Mars: \(T = 1.52^{3/2} = 1.52 \times \sqrt{1.52} = 1.52 \times 1.233 = 1.87\) years. Jupiter: \(T = 5.2 \times \sqrt{5.2} = 5.2 \times 2.28 = 11.9\) years.
The year of Mars is \(1.87\) Earth years and the year of Jupiter \(11.9\) Earth years. Neither \(G\) nor the mass of the Sun was needed.
Saturn orbits the Sun at \(9.5\,\text{AU}\). Use Kepler's third law to find its year, in Earth years.
The filament of a lamp radiates as a black body, \(P = \sigma AT^4\). Its absolute temperature rises from \(2000\,\text{K}\) to \(2400\,\text{K}\). By what factor does the radiated power grow?
The flow through a narrow tube is \(Q \propto Pr^4\), where \(P\) is the pressure difference. An artery narrows to \(90\,\%\) of its radius. By what factor must the pressure difference rise to keep the same flow?
A pendulum is made four times as long and taken to a planet where \(g\) is half the Earth's value. How does its period change? Use \(T = 2\pi\sqrt{l/g}\).
A proportion holds only while every other quantity stays fixed, and only inside the range of the law. \(v = u + at\) is not \(v \propto t\) unless \(u = 0\): a constant added to the law spoils the ratio. Stokes' law, and with it \(v_t \propto r^2\), holds only for small, slow spheres. Temperatures in Stefan's law must be in kelvin. The ratio \(327/27\) in degrees Celsius is meaningless, while \(600/300\) in kelvin is right.
Ratios answer the "what if" questions of every chapter. Kepler's law of periods compares planets. The fourth power of the radius in Poiseuille's law explains a narrowed artery, and merged raindrops fall faster by the square of their radius. Stefan's law compares hot bodies by the fourth power of their temperature. The laws of a vibrating string compare lengths, tensions and masses per length, and the speed of sound compares temperatures and gases. A wire that is stretched, cut or folded changes its resistance by a ratio. The same reasoning gives the extension of a wire and the stopping distance at a higher speed. It also explains the blue of the sky and the force of a hydraulic lift.
- Forgetting the power when a quantity changes.Write the law as \(y \propto x^n\) first. A factor \(k\) in \(x\) gives \(k^n\) in \(y\), not \(k\).
- Using a ratio when the law has an added constant.A ratio works only for \(y = kx^n\). For \(v = u + at\) or a stopping distance with a reaction time, work out both cases in full.
- Mixing units, or using degrees Celsius, in a ratio.Put both values in the same unit, and use kelvin wherever a temperature is raised to a power or divided.