For example let us consider the force one.
$F$ is directly proprtional to $M_1$ and Force is directly proportional to M2,then we say F is directly proprtional to M1*M2,then we remove the proportionality and say (F=GM1M2/r^2)so why do we do this?
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2 Answers
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This is just a little algebra. $F$ is proportional to $m_1$ means $F=k_1 m_1$. Similarly, $F$ is proportional to $m_2$ means $F=k_2 m_2$. So we have $$k_1 m_1 = k_2 m_2$$ $$k_2 = \frac{k_1 m_1}{m_2}$$ then substituting gives $$F = \frac{k_1 m_1}{m_2} m_2 = k \ m_1 m_2$$ where $k=k_1/m_2$. A little additional algebra shows also that $k = k_2/m_1$
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$\begingroup$ I understood that,But what about r²,so can you please derive it with r² too,I mean can You please derive that F=KMm/r² $\endgroup$ Commented May 10, 2021 at 5:43
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When we say $F$ is directly proportional to $M_1$, we mean that keeping other things/factors constant, $F$ will change by as much as $M_1$ changes.
This keeping other things/factors constant part is conveniently omitted that can sometimes cause confusion.
So, if $F$ was directly proportional to $M_1$ and $M_2$, and say, these masses change to $M_1^{'}$ and $M_2^{'}$, getting the new $F^{'}$ is actually a two-step process. First we get $F^{'}_{temp}$:
$$\begin{align}
F^{'}_{temp} &= F.\frac{M_1^{'}}{M_1}
\end{align}$$
In the above transformation, we assumed $M_2$ constant, so $F$ changed only according to how $M_1$ changed.
Now,
$$\begin{align}
F^{'} &= F^{'}_{temp}.\frac{M_2^{'}}{M_2} \\ &= F.\frac{M_1^{'}}{M_1}.\frac{M_2^{'}}{M_2}
\end{align}$$
This time, we assumed $M_1^{'}$ to be constant, so $F^{'}_{temp}$ now changed only according to $M_2$.
This gives:
$$\begin{align}
\frac{F^{'}}{F} &= \frac{M_1^{'}}{M_1}.\frac{M_2^{'}}{M_2} \\
&= \frac{M_1^{'}.M_2^{'}}{M_1.M_2}
\end{align}$$
This is nothing else but saying $F \propto M_1.M_2$.
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$\begingroup$ ,while forming F',why is F'temp multiplied by M2'/M2? $\endgroup$ Commented Jul 15, 2021 at 2:55
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1$\begingroup$ $F^{'}_{temp}$ is just $F$ that has already responded to the change in $M_1$, but is yet to respond to the change in $M_2$. In $F \rightarrow F^{'}_{temp}$, we kept $M_2$ constant, now in $ F^{'}_{temp} \rightarrow F$, we'll keep $M_1$ constant. Finally, since $M_2$ has become $M_2^{'}$, $\frac{M_2^{'}}{M_2}$ is the ratio by which $F^{'}_{temp}$ should change as per the definition of direct proportionality (when all other factors, including $M_1$, are constant). If there was another factor (like $r$), we would need another $F^{'}_{temp\_2}$ before we get $F^{'}$, and so on. $\endgroup$– manisarCommented Jul 15, 2021 at 4:21
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$\begingroup$ yes,thanks a lot,I was not getting the point earlier,a last question,do these responses occur at different instances so that we can differentiate between them? $\endgroup$ Commented Jul 15, 2021 at 10:16
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1$\begingroup$ No, this step-by-step response model is just a trick to understand what's happening. In reality, there is just one response to changes in both $M_1$ and $M_2$. We can use the step-by-step reasoning to understand it because the total response behaves in such a way as if it was a succession of independent individual steps. $M_1$ & $M_2$ affect $F$ individually as if there was no other factor affecting $F$ at any time. This is neat for understanding, but in reality, they are not alone ($M_1$ is accompanied by $M_2$ as a factor & vice versa). So, all this is happening at only one instance of time. $\endgroup$– manisarCommented Jul 15, 2021 at 16:33
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1$\begingroup$ NP. This was a very good question that helped me look back at and understand something very fundamental. $\endgroup$– manisarCommented Jul 15, 2021 at 19:17