Please show all work 5, (20 %) Spring-Mass-Damper System By Newton\'s law, mx eq

Question

Please show all work

5, (20 %) Spring-Mass-Damper System By Newton's law, mx equals the resultant of all the external forces on m inthe figure below in the downward direction To help in determining the signs of the terms in such problems, t is useful to make any assumption concerning the motion: for example, that the mass is moving downward from x-0. In that case the spring is stretched, so spring force kx is upward, and hence oppose downward acceleration. It therefore received a minus sign on the right side of the equation for mx.Since the mass moves down, the damping force cx is upward, and this term must also have a minus sign. The external force f(t) helps downward acceleration, and therefore has a plus sign. The resultant equation is mx =-kx-cx + f(t) Rearranging gives the differential equation of motion in the form mx + cx +kx = f(t). It may have been observed that the effects of gravity do not appear. Why?

Explanation / Answer

when we turn the system upsiode down we will get the same equations irrespec5ive of the gravity using the same method
When we turn the system sideways , is the system is free to move only in the direction perpendicular to gravity, the derivation would again remain the same, but if the sytem is not free to move ohnly along one direction, then the derivation will involve two direction of motion and woiuld be a different and complex one

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