\(F_{drag}=F^{*}_D+F^{*}_c\)
\(F^{*}_D\) is not the resolution of the resultant force \(F+F_{drag}\).
\(F^{*}_D\) the component of \(F_{drag}\) along \(F\), reduces the front force, \(F\) and provide less kinetic energy change in he forward direction. At the same time, \(F^{*}_c\) tends to rotate the particle and impart rotational kinetic energy. In this way, energy change from the original \(F.c\) is split between rotation and forward translation.
It would seem that \(F^{*}_c\) is rotates the particle in opposite directions. \(F^{*}_c\) in the rear rotates upwards and \(F^{*}_c\) in the front rotates downwards. This is actually consistent with a failure scenario when the particle fracture. A bullet, for example, on impact flares up (opens up) at the rear but hold firm at the front (a shiny impact spot).
\(F^{*}_c\) in the rear and \(F^{*}_c\) in the front may not act to rotate the particle in a common rotational sense. The particle is twisted when they rotate in clockwise and anticlockwise direction.
But \(F_{drag}\), as it move through to the rear of the particle, is decreased by the internal forces that holds \(\psi\) together. The rotational forces at the front of the particle dictates the rotation of the particle.
However,
\((F-F^{*}_{D})_{front}=(F-F^{*}_{D})_{rear}\) --- (*)
when the deformed particle has one velocity.
A difference in the front and rear horizontal forces will deform the particle, until (*) holds. As \(F^{*}_D\) in the rear is less than \(F^{*}_D\) in the front, the particle compresses to the front and the rear is flattened. Such a compression adds another force to the equation (*), on an elemental mass \(\delta m\) on the rear surface of the particle.
\((F-F^{*}_{D})_{rear}-F_{material}=(F-F^{*}_{D})_{front}\)
where the right hand side of the equation is of an elemental mass at the tip of the particle, \(F_{material}\) is the force due to material property (compression, elongation).
\(F^{*}_c\) in the rear wobbles the particle and direct the tip of particle away from the horizontal rotational axis. It tends to stretch the particle in the upward direction away from the horizontal rotational axis. \(F^{*}_c\) in front compresses the particle in the upward direction towards the horizontal axis. When the particle is rotating,
\(-F^{*}_c+F_{material}=F^{*}_c\)
\(F_{material}=2F^{*}_c\)
The internal force reverses \(F^{*}_{c}\) at the rear when the whole deformed particle is rotating in one direction; the sense of the rotation being dictated by the force component at the front of the particle.
If you don't want you missile to wobble make sure it is free to deform at the rear so that \(F^{*}_c\) cancels with internal force completely. A soft ass.
In this case, \(F^{*}_c\) in front is solely due the drag force acting through the \(C.G\) of the particle. If the particle first suffers compression under \(F^{*}_{c}\) then,
\(F^{*m}_{c}=F^{*}_{c}-F_{material}\)
we replace \(F^{*}_{c}\) at the front, with the force \(F^{*m}_{c}\) to account for a lower rotation.
If you want your missile not to rotate, it should compress at the front. A soft front and all in all a soft missile. It is after all such material deformations, that the shape of the missile is to fit an aerodynamics profile.
What happen to the end of the particle? The particle is similarly deformed at the rear. The rear is compress towards the tip to account for loss in \(F_{drag}\) passing through the particle. And it is stretched in both vertical directions, up and down, away from the horizontal rotational axis, the result of which is that internal force reverse \(F^{*}_{c}\) at the rear and the particle spins in one direction (as the front).
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Note: \(F_{drag}\) is virtual due to entanglement that shares energy of the particle. \(F_{drag}\) is defined as if the kinetic energy of the particle is changed directly and immediately. The \(C.G\) defines the position of the particle. Moving the \(C.G\) moves the particle. If the the virtual force does not act through the \(C.G\) then relatively, the particle has moved for no apparent reason. Alternatively, if this force does not act through the \(C.G\) it will generate a torque that imparts additional rotational kinetic energy. All energy change should be part of the total energy change that defines this virtual force. Otherwise there must be another force. A force with displacement imparts energy into a system. A virtual force imparts energy directly into a system without displacement.
