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Ok, we all know that an Inertial frame of reference is defined like this:

"An inertial frame of reference is one frame where Newton's First Law holds, therefore, a body has a constant velocity or velocity equal zero. And, the sum of all forces equals zero, there is no aceleration etc..."

But, I always read that in an inertial frame of reference holds $F=ma$, the second law.... It's confuses me... why we can assume $F=ma$ in an inertial frame of reference?

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    $\begingroup$ A couple of things. (A) Keep in mind that the $F$ in Newton's 2nd law is the net force. (B) Consider what Newton's 2nd law says in the case of $F = 0$. $\endgroup$ – dmckee Oct 18 '15 at 16:37
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Ok, we all know that an Inertial frame of reference is defined like this:

"An inertial frame of reference is one frame where Newton's First Law holds, therefore, a body has a constant velocity or velocity equal zero. And, the sum of all forces equals zero, there is no aceleration etc..."

No. An inertial frame is one in which all of Newton's Laws hold. And the first law has content, it tells us that a particle on the top of Norton's Dome that is at rest stays there at rest forever unless disturbed. It's not a practical result since the top of Norton's Dome is unstable anyway.

But it isn't enough to have the first law hold since there are examples of systems where the first law holds and the second doesn't (replace the second law with anything that gives zero acceleration when net force is zero) and examples where the second holds but the first doesn't (e.g. where a particle sits on top of Norton's Dome for an hour and then spontaneously falls down in some direction).

But, I always read that in an inertial frame of reference

People get the first and second law confused and think all kinds of weird things. If you read the originals, the first one tells you about things staying in uniform motion when in uniform motion and having no net force. The second tells you what happens when there is a net force, and they are both talking about kinematics described in an inertial frame. And the third describes the forces themselves and again is about the descriptions in an inertial frame.

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