# Addes mass forces: can a force depend on acceleration?

I always interpreted $F=ma$ as a cause-effect relationship, so I find rather uneasy to accept that the cause can instantaneously depend on the effect.

Is it fine to have a force which depends on an acceleration, in classical mechanics?

I came up with some possible solutions to this:

1. It's perfectly fine for F=m a to be an implicit equation with respect to a.
2. The time derivative of the velocity appears as the result of an approximation of a time-delay.
3. It arises due to assumptions made on the nature of the fluid (i.e. incompressible).
4. None of them
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–  Qmechanic Jun 28 '13 at 20:06

There are two types of problems. a) Forward dynamics where $a=\frac{F}{m}$ or b) Inverse dynamics where $F = m a$.

The $F$ stands for net forces which includes applied forces, reaction forces, and friction forces. Applied forces are generally a function of time, position and velocity. Reaction forces directions are perpendicular to motions so to do no work and friction forces depend on other (interface) forces. So overall you do have cross-dependency on motion and forces in dynamics.

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Is it fine to have a force which depends on an acceleration, in classical mechanics?

EM fields depend on acceleration of their sources. These fields can enter equation of motion of a particle, so indirectly that equation will contain accelerations of other particles. If the particle has non-zero size (sphere), it has parts and can experience self-force due to action of one part on another. In theory, there is such electromagnetic self-force whose one part is proportional to the acceleration of the sphere. Since there is already term $m_0a$ in there, this can be thought of as effective increase (decrease) in inertial mass (Lorentz and Abraham thought about this 100 years ago).

It is not necessary to regard the equation of motion as a cause-effect relationship. The equation $F=ma$ does not suggest such relation at all - both force and acceleration of the particle happen at the same time. We sometimes tend to think of forces as causes and acceleration as effect since we can only detect easily change in velocity after some time, so force seems to happen before acceleration, but in mathematical theory acceleration happens at the same time as the force.

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In classical electromagnetism,we have a force which depends on the acceleration(strictly it depends on its derivative).

Abraham-Lorentz force: It is the recoil force a charge experiences due to the electromagnetic field produced by a accelerating charge.

$$F_{rad} = \frac{\mu_0 q^2}{6 \pi c} \dot{a} = \frac{ q^2}{6 \pi \epsilon_0 c^3} \dot{a}$$

So yes,acceleration dependent forces do exist.

References:

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Abraham - Lorentz force is a formula derived as an approximate expression for EM self-force acting on a charged sphere which was never experimentally confirmed. When taken as exact, it leads to incorrect description of the motion of charged particles. It is not valid to use it to support claim "acceleration dependent force do exist". –  Ján Lalinský Apr 26 at 7:26