A particle of mass m moves in one dimension and is subject toa restoring force w

Question

A particle of mass m moves in one dimension and is subject toa restoring force which is proportional to its displacement x (e.g. a spring – Hooke’slaw) and a damping force which is proportional to its velocity. Derive an expression for the differential equationof the particle’s motion when it is acted upon by a periodic driving forceFocosFt. A particle of mass m moves in one dimension and is subject toa restoring force which is proportional to its displacement x (e.g. a spring – Hooke’slaw) and a damping force which is proportional to its velocity. Derive an expression for the differential equationof the particle’s motion when it is acted upon by a periodic driving forceFocosFt.

Explanation / Answer

   given restoring force propotional to displacement butoppsotily directed = -x where is force constant damping foce is protional to velocitu but oppsitelydirected = -r dx/dt r is damping force per unit velocity external perodice driving force = FcosF t so the total force is equatingis equal to F = md2x/dt2 ( from newton 2nd law    md2x/dt2 = -x +-rdx/dt +F cosF t d2x/dt2 +r/m dx/dt+/m x = F/m cos t d2x/dt2 +2bdx/dt+2x = fcost here 2b = r/m /m = 2 , F/m =f this is differential eq of the motion of theparticle.

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