Imagine the experimental means that would be required to apply a force F0 cos Ωt to a mass...

Imagine the experimental means that would be required to apply a force F0 cos Ωt to a mass. It doesn’t sound so hard if the mass is stationary, but imagine trying to apply such a force to a moving mass! In many physical applications, such as earthquake-induced vibration, the driving force is applied indirectly, by “shaking” the wall, rather than being applied directly to the mass. Specifically, for the system shown in the figure, use Newton’s second law to show that if the wall is

displaced laterally according to δ(t) = δ0 cos Ωt, then the equation of motion of the mass m is mx″ + kx = F0 cos Ωt, where F0 = kδ0. Here, x and δ are measured relative to fixed points in space. NOTE: Observe that such an experiment is more readily performed since it is easier to apply a harmonic displacement δ(t) than a harmonic force; for instance, one could use a slider-crank mechanism (which converts circular motion to harmonic linear motion). Note further that a dis­placement input is precisely what an automobile suspension is subjected to when we drive over a bumpy road.