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I was studying about the derivation of coriolis force and centrifugal force for a particle i.e. the relation between the accelerations of a particle with respect to inertial and uniformly rotating frame of reference in which the z axis and the origin of both frames co incide with each other and after some time i stuck to the following expression-

$$ \bigg(\frac{d^2r}{dt^2}\bigg)_{S_0}=\bigg(\frac{d}{dt}\bigg)_{S}\bigg[\bigg(\frac{dr}{dt}\bigg)_{S}+\omega×r\bigg]+\omega\bigg[\bigg(\frac{dr}{dt}\bigg)_{S}+\omega×r\bigg].$$

In this expression, $$\bigg(\frac{dr}{dt}\bigg)_{S}+\omega×r$$ is the velocity of the particle with respect to Inertial frame so how can we differentiate this quantity with respect to another frame S (which is a rotating frame). I am unable to imagine and visualise this entire equation.

Can anybody help me for its deeper understanding?

Qmechanic
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Utkarsh Kashyap
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  • Actually, the more interesting case to visualize is the case such that an object is at all time subject to the amount of centripetal force that is sufficient to sustain circumnavigating motion. Such a centripetal force does not constrain to circular motion only; it leaves freedom for a form of circumnavigating motion where distance to the center of rotation is oscillating. I have created an interactive animation for that case: Coriolis effect. (There is a companion animation for the case you ask about: centrifugal effect.) – Cleonis Sep 16 '23 at 09:12
  • Related : Velocity in a turning reference frame. – Frobenius Sep 16 '23 at 11:43
  • But I am asking that how can we differentiate one quantity as seen by an Inertial frame with respect to a non-inertial(rotating in this case) frame. Suppose you are at some point on earth between north pole and equator (the shortest distance from your stand point to the north pole and to the equator is same), now you shot a bullet perpendicular to the surface of the earth from your standpoint. From your stand point,it will seem to be moving linearly away from you but from the centre of the earth in an Inertial frame , it will seem to be moving along a conical helix. – Utkarsh Kashyap Sep 17 '23 at 02:58
  • But you will not encounter this trajectory with respect to the earth frame, in earth frame bullet is moving linearly so how can we differentiate the velocity of the trajectory with respect to a frame in which that trajectory is not possible to seen? – Utkarsh Kashyap Sep 17 '23 at 04:00

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