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I was thinking about acceleration due to gravity and I thought of deriving a formula that gives the net acceleration due to gravity between two bodies. Now, by net acceleration, I basically mean the effective acceleration. Please have a look :

Let $A$ and $B$ be two objects with masses $m_1$ and $m_2$ respectively and the distance between them be $d$. Let $F$ be the force of attraction between them and $g_{m_1}$ be the acceleration of $m_1$ due to the gravitational force of $m_2$ and let $g_{m_2}$ be the acceleration of $m_2$ due to the gravitational force of $m_1$.

Now, according ot Newton's Law of Gravitation: $$F = G \dfrac{m_1m_2}{d^2}$$

We know that: $$F = m_1g_{m_1}$$ $$\text{and}$$ $$F = m_2g_{m_2}$$

This implies that: $$m_1g_{m_1} = G \dfrac {m_1m_2}{d^2} \implies g_{m_1} = G \dfrac {m_2}{d^2}$$

In a similar manner: $$g_{m_2} = G \dfrac {m_1}{d^2}$$

Now, this is the part where I think I'm making a mistake.

What I think is that here both the objects are accelerating towards each other with accelerations of $g_{m_1}$ and $g_{m_2}$ respectively.

So, I think the net acceleration between $A$ and $B$ would be $$g_{m_1}+g_{m_2} = G \dfrac{m_2}{d^2} + G \dfrac{m_1}{d^2}$$ $$\ \ = \dfrac {G}{d^2}(m_2+m_1)$$

Now, I think that if this formula is correctly derived, it gives the net acceleration by which two masses mutually attract each other. And wouldn't this formula imply that the acceleration due to gravity does depend on the mass of the object that is into consideration which is actually not the case.

So, that would imply that this formula is wrong. So, please let me know where the error is

Thanks!

PS : All edits on formatting are welcome :)

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  • $\begingroup$ your formula say 2 equal masses don't attract, so it is manifestly wrong. But: what do you mean by "this formula imply that the acceleration due to gravity does depend on the mass of the object that is into consideration which is actually not the case."? $\endgroup$ – JEB May 27 at 14:27
  • $\begingroup$ I have edited my post to get rid of the fact that the earlier formula implied that two objects of equal mass don't attract, it was a misunderstanding of vectors. And what I mean by that statement in quotes is that since the formula involves both of the masses $m_1$ and $m_2$, doesn't that mean that the acceleration due to gravity depends on both of the masses but what truly happens is that acceleration due to gravity is a constant on Earth and does not depend on the mass of the object that is falling down, unlike what my formula suggests. $\endgroup$ – Rajdeep Sindhu May 27 at 14:39
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Your formula is just fine. You can read about the "2 body problem" and solve it entirely. You seem to be upset about that the relative acceleration depends on both masses, but is correct. Think of $m_2$ >>> $m_1$. You get $g_1$ >>> $g_2$. You can aproximate relative acceleration by $g_1$ and the acceleration of $m_1$ towards $m_2$ does not depend on $m_1$.

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  • $\begingroup$ Thanks for answering! What does $m_2 >>> m_1$ and $g_1 >>> g_2$ mean by the way? $\endgroup$ – Rajdeep Sindhu May 27 at 15:20
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    $\begingroup$ It means much larger than $\endgroup$ – Wolphram jonny May 27 at 15:44
  • $\begingroup$ Okay, thanks! That clears it :) $\endgroup$ – Rajdeep Sindhu May 27 at 16:56
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Acceleration is a vector (see the edit for more on vectors). Thus, in order to find the relative acceleration between two objects, you need to do a vector subtraction. However, since the gravitational force between two objects initially at rest would always lie on the same line, i.e. the one connecting the two objects, you can do away with using vectors in their full glory but you'd still need to keep track of signs. Let's say you take the direction going from $1$ to $2$ to be positive then the other direction would be negative. Thus, if $g_1$ is $Gm_2/r^2$ then $g_2$ would be $-Gm_1/r^2$. As you can see, this resolves your issue.

Edit

A vector is a quantity with both a magnitude and a direction. For example, in order to characterize the motion of a particle, you need to say how fast it is moving but also which direction it is moving. Accelerations are also the same. You need to say how much something is accelerating but also in which direction is the acceleration pointing. Now, for gravitational forces between two particles which were initially at rest, the forces are always on the same line. So the direction is specified simply by a plus or a minus sign. If plus means from $1$ to $2$ then minus means from $2$ to $1$. You see, since the gravitational forces on the two bodies would be in the opposite directions, the accelerations produced would also be in opposite directions. So you need to take into account that fact (using the plus or minus signs) when adding or subtracting these accelerations.

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  • $\begingroup$ Thanks, but I don't know much about vectors. And I'm sorry, but I also fail to see how this resolves my issue, please elaborate if possible... $\endgroup$ – Rajdeep Sindhu May 27 at 14:12
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    $\begingroup$ @RajdeepSindhu I've added a paragraph, hope it helps. If not, you should try to see the Wikipedia page of vectors or read up about them from a nice book (if you're in highschool, you can pick up Halliday/Resnick or HC Verma), it's a simple and interesting topic. :) $\endgroup$ – Dvij D.C. May 27 at 14:18
  • $\begingroup$ Thanks for the addition and fortunately, I just bought those two books a while ago. But, I'm still not clear. Acceleration is a vector and in this case, the accelerations are in opposite directions, so I have subtracted them, just as you said. I think I don't get what you mean by taking the plus or minus sign in account. Are you referring to this symbol : $\pm$ or just saying that i should take care of the directions as acceleration is a vector quantity. Pardon me if I'm being annoying. $\endgroup$ – Rajdeep Sindhu May 27 at 14:27
  • $\begingroup$ @RajdeepSindhu Ah, I see your confusion. So you added the minus sign because you do understand that the accelerations are in opposite direction? If yes, I'll clarify my answer more by the night. Let me know. $\endgroup$ – Dvij D.C. May 27 at 14:28
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    $\begingroup$ I see what I did. I should have put a plus sign there. Silly me, feels funny after you realize it. $\endgroup$ – Rajdeep Sindhu May 27 at 14:30
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I see what you have calculated is called relative acceleration. Please go through vectors to understand how force and acceleration works.

Now, you show us that $g_{m_2}$and $g_{m_1}$ doesn't depend on masses of the corresponding objects you are looking at, where as $g_{m_2}+g_{m_1}$ does.

First let me tell you what relative motion is, ride a bike and record the speed of another vehicle moving beside you with the same speed. You'll find it at rest as if it's not moving at all, and the same goes for a vehicle travelling in opposite directions but here the speeds will sum up.

That's what you did while calculating "what you call the total acceleration (should be relative acceleration)".

When you calculate the relative acceleration you shift your point of observation and that's why you see is mass $m_2$ moving with acceleration $g_{m_2}+g_{m_1}$wrt to $m_1$. And that hence the $m_2$+$m_1$ pops up.

Look for relative motion and vector you'll clear yourself.

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  • $\begingroup$ Thanks a lot! ${}$ $\endgroup$ – Rajdeep Sindhu May 27 at 16:55

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