Question

A particle of mass m is subject to a damping force equal to -bv, and also a driving force equal to Fa coswat, but there is no spring force. Find a solution for x(t) by guessing one of the form x(t) = A cos watB sinwat. If you write your solution for x(t) in the form of C cos(wat- ), what are C and ø? (This problem is a special case of the one we did in Lecture 20, but redo it from scratch. You can check various intermediate results with the results from class.)

Answer 1

Ans: The equeten o mohon o09 clampeal daun oscallatur rs givem by whuse -bu= bd + b da Ka = Fib) dt2 fs a damping force and fit) dr exolermat diving OuT 9IVn pro blem, thoae no Spring foree furce 9an bda FCt) व६ Fit)=FaCos (04t) md2oc + dt 2 - Fa Cos(wat) we locgin oextemal dnving foree= Fit) = Fde wIth Fa aeal SO forer rs acal past ot this So, actual dof ving fa cos4 we ape toying to solve he ean md2o 4 bdo Fdelwas we'll try cmplkeo funchion (tto) Shore CE acal number Amplikide act)= Ce and we have added the p adJu Ska ble phase factor iwat te) 1s q solun to the eauaHon TT XC= ce wIh the comp oo diying Foree Fd elt &cos(wat +0), wil beq solaton to te Its real paot ean with oeal Physical dafy ing force, Fd cos(wat) in eah i plug, 2t) cewt+e) i(wat+o) we get - mu2ce + ibw ce Fol e fwit

Nao, canalling out He factor efwd b ele to other sidle, 1,D and Shift -mu2c.el+ rbcee Fel Fdele Fdel6 -muffbWd C = C € aeal number s01 w hats 9oing on, and 01ght Sice 10ok6 Vike compler Th can be unlangu by digramminy he compler no in demo minaler b put, the eauahan, mwd2t fbwd in Ta nd ibuy ntation CP Fd icetp) Fd e mu2 The Complex number -mw+ fbw -mwa2t fbud=ep -i(e+0) e So will be ocal 9 fs deal so fa C = alt)= e (wat- 7 m2 wa4 +(buy) bod tant Heae and mwa tan 1e mwd w46-p) with we hare JuSt establishd hat Fd nence C= tant-b and P Jm2wd +ibwa) sulun to dfun dampe osscilalor ean. ean ( Ts a

Fa elut ->wIm comP dnving er Fdeab so eauetting. eat parh oftuo sids ot ean Sinu Fd Cos ud b STh oea dnving fenee solun and ptanm Fd whae -b C = Jnwat(bu)