The International System of Units — Question 10

Question 10

Does the SI fix every constant in nature?

The SI is defined by fixing seven numerical values: the specified caesium frequency $\Delta\nu_{\mathrm{Cs}}$, vacuum light speed $c$, Planck constant $h$, elementary charge $e$, Boltzmann constant $k$, Avogadro constant $N_{\mathrm A}$, and the luminous efficacy $K_{\mathrm{cd}}$ for the specified monochromatic radiation.

For this problem use the supplied relations $$E=h\nu,\qquad E_{\mathrm{th}}=kT,\qquad W=mg.$$ Here $E$ is a photon’s energy, $\nu$ its frequency, $E_{\mathrm{th}}$ a defined thermal energy scale (not a body’s total energy), $T$ thermodynamic temperature, $W$ an object’s weight force, $m$ its mass, and $g$ local free-fall acceleration. These relations are supplied; no quantum or thermal theory is required.

A report makes three claims:

A. Since $h$ and $k$ are exact, every photon energy and every temperature is exact.
B. Since $\mathrm{m}$ and $\mathrm{s}$ are defined exactly, $g$ is exact and identical at all sites.
C. If two sites use the same kilogram, an object must have the same weight at both.

1. Assess A. If $\nu$ is measured, where can uncertainty in $E$ enter? If $E_{\mathrm{th}}$ is measured, where can uncertainty in $T=E_{\mathrm{th}}/k$ enter?

2. Assess B and C. For one object of fixed mass at sites with $g_A=9.80\ \mathrm{m\,s^{-2}}$ and $g_B=9.75\ \mathrm{m\,s^{-2}}$, find $W_B/W_A$ and the percentage decrease in weight. Treat these values as exact for this comparison.

3. Write a replacement statement explaining what exact unit definitions do guarantee, and what they do not guarantee about physical measurements or location-dependent quantities.

Question 10 - Solution

Does the SI fix every constant in nature?

(a) Exact proportionality constants do not fix every input.

$$\delta E=h\delta\nu,\qquad\delta T=\frac{\delta E_{\mathrm{th}}}{k},\qquad\delta h=\delta k=0.$$

Claim A is false. In $E=h\nu$, an uncertain measured frequency gives an uncertain inferred photon energy even though $h$ is exact. Likewise $T=E_{\mathrm{th}}/k$ inherits the uncertainty in the measured thermal energy scale. Dividing or multiplying by an exact constant contributes no new uncertainty, but does not erase the input’s uncertainty.

The SI fixes selected numerical values to define units. It does not assign exact values to every photon frequency, energy, temperature, or other physical quantity.

(b) A shared unit does not require identical local physics.

Claim B is false: $g$ is not one of the seven defining constants. Its having unit $\mathrm{m\,s^{-2}}$ specifies how to express it, not its value or its dependence on location. Exact definitions of metre and second do not make every quantity expressed using them exact.

Claim C confuses mass and weight. With the same fixed mass in both supplied models, $$\boxed{\frac{W_B}{W_A}=\frac{mg_B}{mg_A}=\frac{9.75}{9.80} =\frac{195}{196}\approx0.994898}.$$ The decrease relative to site A is $$\frac{W_A-W_B}{W_A}\times100\% =\frac{1}{196}\times100\%\approx\boxed{0.5102\%}.$$ The kilogram is shared and the object’s mass is unchanged in the stated model; the local weight force differs.

(c) A defensible replacement statement.

“The SI’s defining constants establish a common system of units through fixed numerical values. The same unit definitions can be realized and used at different sites. Practical realizations and measurements still have uncertainty, and a physical quantity may vary with conditions or location even when everyone uses the same units.”

Improved measurements can refine a value such as local $g$ without changing the SI. Agreement on units makes comparisons meaningful; it does not require equal results for different physical conditions.