I just have a question about relationships between physical quantities. Is it true that if:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] z \propto x \ \textrm{and}\ \ z \propto y [/tex]

then [itex] z [/itex] will always be in the form:

[tex] z = Cxy [/tex]

Where [itex] C [/itex] is a constant?

What prompted me to ask was Newton's Universal Law of Gravitation. In the textbook I have, it wastes no time arriving at the typical equation by simply stating that Newton determined that the gravitational force between two masses was directly proportional to theproductof those masses and inversely proportional to the square of the distance between them. So my question could be restated: In the expression:

[tex] F = G\frac{m_{1}m_{2}}{r^2} [/tex]

Is the fact thatFis proportional to theproductof the masses a necessary consequence of the mathematics? I.e., does the math demand that it be so in order to satisfy the condition that the gravitational force is directly proportional to (varies linearly with) m_{1}andalsovaries linearly with m_{2}?

Or is there an alternative (non-physical) way of formulating the expression that still satisfies the stated condition. For instance, would:

[tex] F \propto m_1 + m_2 [/tex]

fit the bill mathematically? It seems to me that when you consider the relationship between F andeach oneof the masses, all other things being equal, it is still linear.

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# Law of Gravitation

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