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Below is a list of some measurable effects that emerge from circular motion, but in fact are not physically real: 1) Reactance in ac circuits from oscillating periodic motion. It impedes the flow of ac current, but is not a physical property of the circuit element producing it like resistance is. 2) Time dilation from the circular flow of time; by making the speed of light invariant, you are saying that the total flow of motion through spacetime is c, whether it's motion in the time dimension, the spatial dimensions, or both; this total flow of motion is the circular flow of time following spacetime curvature. 3) The effect in lump ac circuits that allows you to view the sinusoidal source as instantaneously traveling through the circuit—even though the wavelength is much longer than the closed loop circuit path it travels. 4) The outward push that an object in circular motion experiences which is mistakenly taken as the centrifugal force; this is an effect from the momentum of the ...
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Now let's look at the acceleration function for circular motion, which is the second derivative of the rotational function with respect to time: F(t)ᵣₒₜ = Re^(iwt) F'(t)ᵥₑₗ = iwRe^(iwt) F''(t)ₐ = -w²Re^(iwt) The units here are R/w² which is dist/sec² for acceleration. We notice that the function is being multiplied by the -1 operator. Which means the acceleration vector is opposite the original rotational vector on the real axis. When we take the tail end of the acceleration vector and attach it to the front tip of the rotational vector; we notice that the acceleration points inward towards center as the rotational vector traces a circle. This then is the centripetal acceleration which acts on an object in circular motion, keeping it from traveling in a straight line tangent to the circle. Because of Newton's third law of reactionary forces, the centripetal acceleration causes an equal and opposite acceleration called the centrifugal acceleration which is a real for...
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Page 5 of 6

The imaginary unit i=√-1, is not a real number, it is a math construct that intertwines with real numbers in the complex number plane. This intertwining produces real measurable effects in the physical world without in fact being real. The first derivative of the exponential function in Euler's formula reveals that circular motion is encoded with the imaginary unit i, here it represents a direction perpendicular to the circle's radius at every point on the circle. So linear motion is actually the limiting case of circular motion. Where you get piecewise linearity for very small sections of the circle(sect→0). And linearity over larger sections as the radius increases(r→∞). This is why the kinetic energy formula for linear motion applies to circular motion as well: Krot = ½Iw² Where: w is angular velocity and I is moment of inertia  A point mass m at radius r has a moment of inertia(I): I = mr² and v = wr Krot = ½mr²w² = ½mv² By keeping the encoded imaginary unit i, we get: K...
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