lect_07 - Total Differentiation of a Vector in a Rotating...

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1 Total Differentiation of a Vector in a Rotating Frame of Reference Before we can write Newton’s second law of motion for a reference frame rotating with the earth, we need to develop a relationship between the total derivative of a vector in an inertial reference frame and the corresponding derivative in a rotating system. k ˆ A j ˆ A i ˆ A A z y x + + = r in an inertial frame of reference, and k ˆ A j ˆ A i ˆ A A z y x + + = r in a rotating frame of reference. Let be an arbitrary vector with Cartesian components A r k ˆ A j ˆ A i ˆ A A z y x + + = r in an inertial frame of reference, then If + + + + + = dt dA k dt k d A dt dA j dt j d A dt dA i dt i d A dt A d z z y y x x ˆ ˆ ˆ ˆ ˆ ˆ r Since the coordinate axes are in an inertial frame of reference, 0 ˆ ˆ ˆ = = = dt k d dt j d dt i d + + + + + = dt dA k dt k d A dt dA j dt j d A dt dA i dt i d A dt A d z z y y x x ˆ ˆ ˆ ˆ ˆ ˆ r k dt dA j dt dA i dt dA dt A d z y x ˆ ˆ ˆ + + = r (Eq. 1)
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