PROBLEMS TO BE WRITTEN UP AND TURNED IN AT RECITATION

     These problems are due at the beginning of Test 4 on Tue 5/3.  The
     test bags will be in their usual positions.  Please put your HW in
     the test bag for your recitation as you enter the classroom.  Thanks.
     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.

         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
              Wolfson.  Show that the sum of the cos((omegaD)t) terms = 0 
              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) 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).