Understanding Fictitious Forces: A Mathematical Derivation

[putongren]

I'm having trouble understanding fictitious forces, so I looked up the wiki for it http://en.wikipedia.org/wiki/Fictitious_force" . In the wiki entry, I have trouble understanding the mathematical derivation for fictitious forces in the early part of the article (along the mathematical derivation section, there's a mathematical model with a particle, an inertial frame of reference A, and a non-inertial frame of reference B, with displacement vectors written between those entities.) The equations for displacement vector between A, B, and the particle is given. Then the equations are differentiated twice to get the acceleration vector and multiplied by mass to get the force vector. The part I do not understand is the step where the author of the article writes Fb = Fa + F fictitious. I sort of understand how the author derived this mathematical statement, but I do not understand the meaning and implication of it. Does F fictitious represents Xab differentiated twice and multiplied by mass? If so, does that mean the fictitious force is actually force vector between inertial frame of reference A and non-inertial frame of reference B? I thought that the fictitious force is responsible for acceleration of mass in a non-inertial frame of reference. Please help.

 

[Valerie Park]

The equation Fb = Fa + F fictitious is a way of expressing the fact that the net force acting upon a particle in a non-inertial frame of reference is equal to the sum of the force vector between an inertial frame of reference A and the particle plus a fictitious force vector. The fictitious force vector is a vector that does not actually exist in the physical world, but rather is a mathematical construct used to explain why objects experience acceleration when moving from one frame of reference to another. The fictitious force vector can be thought of as representing the acceleration experienced by the particle due to the change in frames of reference.

 

[charng]

I can understand your confusion with the mathematical derivation of fictitious forces. Let me try to explain it in simpler terms.

Fictitious forces are forces that appear to act on an object in a non-inertial frame of reference, but are actually due to the motion of the reference frame itself. They are not real forces, but rather a mathematical construct used to simplify the equations of motion in a non-inertial frame.

In the derivation you mentioned, the displacement vectors between the inertial frame A, non-inertial frame B, and the particle are given. These vectors represent the position of the particle in each frame. When these vectors are differentiated twice, we get the acceleration vector of the particle in each frame. This is where the fictitious force comes in.

The author writes Fb = Fa + F fictitious, where Fb is the force on the particle in the non-inertial frame B, Fa is the force on the particle in the inertial frame A, and F fictitious is the fictitious force. This equation simply means that the total force on the particle in the non-inertial frame is equal to the sum of the real force in the inertial frame and the fictitious force.

Now, to answer your question, F fictitious does represent Xab differentiated twice and multiplied by mass. And yes, it represents the force between the two frames, but it is not an actual force acting on the particle. It is simply a mathematical construct used to account for the acceleration of the particle in the non-inertial frame.

I hope this explanation helps you understand the meaning and implication of the mathematical derivation of fictitious forces. Keep in mind that these forces are not real, but rather a mathematical tool to simplify calculations in non-inertial frames.