PROBLEMS TO BE WRITTEN UP AND TURNED IN AT RECITATION

     These problems are purely for extra credit; i.e. if you have not
     already collected your 1700 HW points.  However, they are instructive;
     they are similar to G1 and G2 (although the algebra and trig is a bit 
     more involved); also, they show you the origin of our expressions for
     the phase constant and the impedance in a series LRC AC circuit. These 
     problems may be turned in at your final recitation meeting or at the 
     beginning of Test 5 on Mon 4/30.  The test bags will be in their usual 
     positions.  You may put your HW in the test bag for your recitation as 
     you enter the classroom.  Each problem should be written on a single 
     sheet of paper (front and back), or if multiple sheets are used, staple 
     them together.  For full credit, the work must be neat, with clearly 
     labeled diagrams (no diagrams necessary for these problems, they are 
     purely math problems).  The labels must be in BOTH words and symbols.  
     For each part of the problem, you must BRIEFLY explain your strategy in 
     words, as well as neatly showing your physics equations and math, with 
     proper units. 


         G3 and G4:  (10 points each)

         The differential equation for a driven LRC circuit is

           (q/C) + R(dq/dt) + L(d^2q/dt^2) = (EMFm)sin((omegaD)t)  Eq (2)

         where (EMFm)sin((omegaD)t) is the driving EMF as a function of time,
         EMFm is the amplitude of that driving EMF, and omegaD is the driving
         angular frequency.  Your task is to show that

           q(t) = -(Q_m)cos((omegaD)t - phi)

         is a solution of this differential equation, thereby finding the
         constants Q_m and phi (and, in the process, the impedance).

         G3.  Substitution of q(t) in Eq (2) gives terms in sin((omegaD)t) 
              and terms in cos((omegaD)t).  HINT: Use the trig identities for 
              sin(alpha+-beta) and cos(alpha+-beta) as found on page A3 of
              Young&Freedman.  Show that the sum of the cos((omegaD)t) 
              terms = 0 if and only if

                                tan(phi) = (1/R)(X_L-X_C)

              where X_L and X_C are the inductive and capacitive reactances.

         G4.  (a) Show that the sum of the sin((omegaD)t) terms =
                  (EMFm)sin((omegaD)t) if and only if

                              EMFm/(omegaD)Q_m = R/cos(phi)

                   and therefore solving for Q_m while at the same time
                   showing that Z = R/cos(phi), where Z is the impedance.
                   HINT: Use sin^2(phi) + cos^2(phi) = 1.

              (b) From (a), show that Z = sqrt((X_L-X_C)^2+R^2).
                  HINT: sec^2(phi) = 1 + tan^2(phi).