The following argumentation has been used in the good old days to teach us about the ideal gas laws of Boyle and Gay Lussac.
- Experiments reveal that the pressure $p$ is proportional to the amount of gas $n$
if temperature $T$ and volume $V$ are held constant. Thus: $\, p \sim n$ .
- Experiments reveal that the pressure $p$ is inversely proportional to the volume $V$
if temperature $T$ and amount of gas $n$ are held constant. Thus: $\, p \sim 1/V$ .
- Experiments reveal that the pressure $p$ is proportional to the temperature $T$
if volume $V$ and amount of gas $n$ are held constant. Thus: $\, p \sim T$ .
- Experiments reveal that the volume $V$ is proportional to the temperature $T$
if pressure $p$ and amount of gas $n$ are held constant. Thus: $\, V \sim T$ .
Then the book (*) tells us that from (2) it follows that $p_1\cdot V_1 = p_2\cdot V_2$ : Boyle's law.
A next step is to introduce absolute temperature $T$ (Kelvin) instead of relative temperature.
Then from (3) we have $p_1/T_1 = p_2/T_2$ : pressure law of Gay Lussac.
The book proceeds with (4) and the volume law of Gay Lussac : $V_1/T_1 = V_2/T_2$.
And then comes a clue. What happens if we change the volume $V$ as well as the temperature $T$ ?
The book says that we should do this in two subsequent steps:
- Keep the volume $V$ constant and raise the temperature $T$.
Then we have according to Gay Lussac:
$\;p_1/T_1 = p_2'/T_2 \quad \Longrightarrow \quad p_2' = p_1\cdot T_2/T_1$
- Keep the temperature $T$ constant and change the volume $V$.
Then we have according to Boyle:
$\;p_2'\cdot V_1 = p_2\cdot V_2 \quad \Longrightarrow \quad p_1\cdot T_2/T_1 \cdot V_1 = p_2\cdot V_2$
It follows that:
$$
\frac{p_1 \cdot V_1}{T_1} = \frac{p_2 \cdot V_2}{T_2} \quad \Longrightarrow \quad p\cdot V = C\cdot T
$$
Where $C$ is a constant that depends only on the amount of gas $n$.
It is supposed that the same procedure may be applied for Newton's law of gravitation:
- Experiments reveal that the force $F$ is proportional to one of the two masses $m$
if the other mass $M$ as well as the distance $r$ between the masses is held constant: $\, F \sim m$ .
- The same argument hold for the other mass if mass $m$ and distance $r$ are held constant.
Thus: $\, F \sim M$ .
- Experiments reveal that the force $F$ is inversely proportional to the distance $r$ squared
if the mass $M$ as well as the mass $m$ are held constant. Thus: $\, F \sim 1/r^2$ .
Now we are going to derive Newton's law in three steps.
(Don't get confused by the naming!)
- Keep the distance constant and change only the mass $m$ :
$
F_1/m_1 = F_2/m_4 \quad \Longrightarrow \quad F_2 = F_1 m_4/m_1
$
- Keep the distance constant and change only the mass $M$ :
$
F_2/M_1 = F_3/M_4 \quad \Longrightarrow \quad F_3 = F_1 m_4 M_4/(m_1 M_1)
$
- Keep the masses constant and change only the distance $r$ :
$
F_3 r_1^2 = F_4 r_4^2 \quad \Longrightarrow \quad F_4 = F_1 m_4 M_4 r_1^2/(m_1 M_1 r_4^2)
$
It follows that:
$$
\frac{F_4 r_4^2}{m_4 M_4} = \frac{F_1 r_1^2}{m_1 M_1}
\quad \Longrightarrow \quad F = \mbox{constant} \times \frac{m M}{r^2}
$$
(*) Reference:
DR. SCHWEERS EN DRS. P. VAN VIANEN
NATUURKUNDE
op corpusculaire grondslag
DEEL I
voor de onderbouw van het v.h.m.o.
ZEVENDE DRUK (1960)
L.C.G. MALMBERG 's-HERTOGENBOSCH