Fundamental Reason for Moving Frame Effects

There is a great deal of interest on the blog in our group discussion on moving frame effects. The fundamental reason for them is that the unit vectors of a moving frame are themselves moving in time. If the unit vectors are denoted e sub i, i = 1, 2, 3 then

d e sub i / dt = omega x e sub i

where omega is the angular velocity vector. The cartesian unit vectors i, j and k are not moving in time. This is the theory developed by Newton. He was aware of centripetal acceleration first defined by Huygens in 1659, but did not develop the correct theory. The force on the centre of mass of a gyro in the Cartesian frame is:

F = mg k

where g is the acceleration due to gravity (g = – mMG / R squared, where R is the radius of the earth and M is the mass of the earth). A spinning top that is not spinning, when tilted over, would simply fall over in the Cartesian frame (often called the Newtonian or inertial theory). When it is spun, the equation of motion is:

F = mg k = m (dv / dt + omega x v) (moving frame)

The movement of the centre of mass of the gyro is intricate, as everyone knows. However F = mg k still applies and the top cannot rise above the ground. If another lifting force is applied it can of course rise above the ground. This is not what is meant by counter gravitation, however. The latter term applies to decreasing g itself (UFT318 – UFT320). I was interviewed by BBC Radio Three a few years ago on counter gravitation and have broadcast a few essays on the subject. ECE2 is able to describe counter gravitation. The first correct theory of the gyro was developed in the late eighteenth century, by Euler and Lagrange in the seventeen fifties, the Euler Lagrange equations used in many UFT papers. Euler was born in Basel and became Director of the Berlin Academy, followed by Lagrange. A complete understanding of the centrifugal and Coriolis accelerations was not achieved until 1835 by G. G. Coriolis. They are due to the expression for acceleration in a moving frame.

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