Could a sine or cosine functions describe the motion of a ferris wheel, if the center of the wheel moved up and down during the period?
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As stated the answer simply cannot be determined. The answer to the one that I am guessing you meant is: not in general.
The height h of a point from the ground on a ferris wheel of radius R and angular speed and center height H can be modeled by
If the height of the center H is now varied sinusoidally, then
which is the sum of two sinusoidal functions. The only way that this is going to be a sinusoidal function is under the following conditions:
(oscillations are perfectly in phase)
(oscillations are perfectly out of phase.)