Differences between this example and the original are given as bold italics.
In the system below a mass, m, is hung from a rectangular frame by a spring (k). There is viscous friction between the frame and the mass on either side (b). The distance xin positive upwards) is measured from a fixed reference and defines the position of the frame. The distance y (measured downwards) is the position of the mass relative to a fixed reference.
Let's draw a free body diagram of the reworked system. Again, we only need one free body diagram becasue the position xin is known; only y is unknown. There are then four forces in the free body diagram:
These are shown below, along with the resulting equations of motion.
For comparison the original system is shown below.
Examing the two depictions of the system we can clearly see that z=xin+y (if xin increases and y remains constant, z increases; if y increases while xin remains constant, z also increases), or y=z-xin. Replacing y by z-xin by the result above yields the original result.
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