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Is $F=ma$ a general formula for all cases in the physical world? or Is it only limited in linear motion?

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$F=ma$ is a general formula. What I mean by general formula is that it is true for both translational ("linear") and rotational motion.

However $F$ doesn't correspond to any force acting on a body. It actually refers to the net force acting on the body.
Example: Say a ball is rolling down an inclined plane. $F$ corresponds to combination of the gravitational force, reaction force from the surface of the plane, air drag, friction force between the ball and the plane.

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In short $F = ma$ is a law and defines the outcomes of the interaction with surrounding, whereas $F = -k x$ is a function which defines how a force varies with time. For example (say) $F= 6\ \mathrm N$ but here too $F=ma$ is true and defines how a body would move under the action of a force of given magnitude (here $6\ \mathrm N$).

You may also like to look at it with the following perspective:

When we say $v = \dfrac {\mathrm dx}{\mathrm dt}$ we mean that at any given instant $v$ is defined as the derivative of $x$ but it does not say anything about how it varies with time (unless you know what $x$ is as a function of time), whereas $v = a x$ tells us how the magnitude of $ v $ varies with time.

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$\vec{F}=m\vec{a}$ is the general formula. Written in its vectorial form it aplies for translational motion of any kind, not just linear.

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Well, The second law states that the acceleration of an object is dependent upon two variables - the net force acting upon the object and the mass of the object.

If it's written in an equation, then it states: a = sigma F/m

a and F are vectors, meaning that the magnitude and direction can vary, so they don't need to be linear motion

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