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CH3 - Accel frame

# CH3 - Accel frame - CHAPTER 3 MOTION OBSERVED IN A ROTATING...

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1 CHAPTER 3 MOTION OBSERVED IN A ROTATING FRAME 3.1 Pseudo-force in the Rotating Frame Consider that we have an inertia frame S with coordinate system xyz fixed to it and another frame S' (with coordinate system x'y'z' attached to it). S' is rotating with a constant angular velocity with respect to the fixed coordinate system xyz. Suppose that there is a vector , which is a function of the time, in the space. It is very clear that the time derivative of this vector function observed from the two frames S and S' will be different because the frame S' is rotating with respect to S. For example, if is a constant vector observed from S (i.e. ), then the time derivative of observed will not be a zero vector or (here we use to represent the rate of change of the vector observed from the rotating frame S'). In the next paragraph, we will try to find the relation between and . In the inertia frame, the vector can be written in the component form by using the xyz coordinate system: . (3.1) Likewise, in the rotating frame S', the same vector is written in the component form by using the x'y'z' coordinate system, which is attached to the rotating frame S' itself: . (3.2) Now, if we consider the rate of change (or the derivative) of the vector in the frame S, then: y x z z' x' y'

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2 (3.3) and similarly, we can also find the rate of change of the vector observed in the rotating frame S': (3.4) Notice that even in the rotating frame S', the rate of change of the unit vectors , and observed from the S' frame is zero because the x'y'z' coordinate system is rotating together with the S' frame. In order to find the relation and , we can write: (3.5) Now, the unit is indeed also rotating with respect to the inertia frame S with an angular velocity of as it is attached to the rotating frame S'. The term is indeed the velocity of the rotating unit vector observed from the inertia frame S. From section 3, we notice that: . Similarly, we also have: and . Substitute back to the equation, we have: (3.6)
3 That is to say, if we have a vector in the space, the rate of change of the vector observed in the inertia frame S is different to that observed in the non-inertia rotating frame S' and is related by equation (3.6).

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