- #1
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I just have a question about relationships between physical quantities. Is it true that if:
[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 the product of 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 that F is proportional to the product of 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) m1 and also varies linearly with m2?
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 and each one of the masses, all other things being equal, it is still linear.
[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 the product of 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 that F is proportional to the product of 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) m1 and also varies linearly with m2?
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 and each one of the masses, all other things being equal, it is still linear.