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Once one has learned of Electromagnetic force, they see that the amount of force experienced by particle given a field is proportional to the particles electric charge.

Similar situation the gravitational case where one sees that the gravitational force experienced by a particle is proportional to its mass.

But, if one sees the other side of the equation they see $ma$ for both cases, and further this $m$ is the exact same as the mass controlling size of gravitational force. Is there any deep reason why one side of Newton's second law ended up being $ma$ instead of say $qa $? ($ q$ is charge)

It seems strange to me because we say gravity and electromagnetism are both fundamental forces but the 'factor' for gravity controls the actual acceleration caused on any particle by any type of force. Essentially, I am confused why mass seems to be more fundamental to calculating acceleration than charge.

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    $\begingroup$ Does this answer your question? Is there a fundamental reason why gravitational mass is the same as inertial mass? $\endgroup$
    – Solomon Slow
    Commented May 18, 2022 at 0:00
  • $\begingroup$ Tangentially related I guess $\endgroup$
    – Brian
    Commented May 18, 2022 at 0:04
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    $\begingroup$ @Aplateofmomos Can you explain why it's not precisely the answer you're looking for? The $m$ which appears in Newton's 2nd law is called the inertial mass, while the $m$ which appears in Newton's law of gravity is called the gravitational mass. The fact that they are the same (or at least proportional) is a strange mystery in the context of Newtonian physics, but is explained beautifully in the context of general relativity. $\endgroup$
    – J. Murray
    Commented May 18, 2022 at 3:30
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    $\begingroup$ @Aplateofmomos If this indeed does not capture the essence of your question, some clarification would go a long way toward helping somebody write a good answer for you. $\endgroup$
    – J. Murray
    Commented May 18, 2022 at 3:31
  • $\begingroup$ this answer of mine might be relevant physics.stackexchange.com/questions/698224/… $\endgroup$
    – anna v
    Commented May 18, 2022 at 4:58

2 Answers 2

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Newton's second law states that the force experienced by a body is vectorially equal to the rate of change of its momentum which is defined as mv. For a fixed mass, this rate reduces to the familiar form of F=ma. This explains the presence of mass on the r.h.s. of the equation. The most fundamental mechanical law (Newton's second law) relates any force with momentum which is defined using mass. The left hand side of the equation is the definition of force for the specific system and so it takes on different forms - using charge for electrostatic, current for magnetostatic, mass for gravitational and so on. So while the l.h.s. is defining the nature of the force acting on your chosen system, the r.h.s. is defining the standard mechanical effect of any force, which is to produce acceleration in an object of fixed mass. Does this answer your question?

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Mass requires input of energy to accelerate...Newton's second law is a description of the changes that a force can produce on the motion of a body, not what the body adheres to on the substantive force

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