Histogram of the student score distribution

The full grading rubric for midterm

Part 1a (34 points) : Plot of -MB cos (theta)

   100 %: practically perfect
    80 %: the value of theta max is wrong (pi / 2)
    75 %: the value of theta for max potential is not indicated
    60 %: shifted by pi/2
    60 %: cos theta is plotted
    50 %: sign error in formula, min OK, max not at pi, but at pi/2
    40 %: plot is wrong, except for the fact that there is a minimum at
          theta = 0
    40 %: sign error in U and jagged (not smooth) graph of U with incorrect
          max positions
    33 %: - sin theta is plotted
    33 %: theta = 0 is not min, but max, and other extrma positions are
          incorrect
    33 %: plotted something like sin (theta), but with incorrect
          periodicity
    10 %: some discussion is presented, but no clear direction is indicated
     0 %: answer missing, or completely incorrect

Part 1b (33 points) : cos (theta) ~= 1 - theta^2 / 2, and so a SHO, since U
                      \propto \theta^2 and K \propto \dot \theta^2.

   100 %: practically perfect
    95 %: would have been perfect, were it not for (numerical) mistakes
          accumulated from previous part(s)
    85 %: correct except that the sign is incorrect
    75 %: small angle approximation was used incorrectly (sin theta = \dot
          theta)
    70 %: equation of motion derived, but small angle approximation is not
          used.
    50 %: qualitative discussion only (or only the first  derivative is
          discussed)
    50 %: correct, or nearly correct, SHO EOM is shown, but it is not
          derived correctly
    33 %: U = - MB cos \theta but no correct discussion from it
    33 %: U \propto sin theta =~ theta has been used
    25 %: correct starting formula, but no or little development from it
    10 %: some discussion is presented, but no clear direction is indicated
     0 %: answer missing, or completely incorrect

Part 1c (33 points) : K = I \dot \theta^2 / 2, U = M B \theta^2 / 2, and so
                      w = \sqrt(MB/I)

   100 %: practically perfect
    90 %: correct except for a (small) numerical error
    75 %: correct except m is used in place of I
    60 %: dimension error
    50 %: I = M R^2 (not correct for the compass even if M is mass) is used
          to cancel M (magnetic moment)!
    50 %: dimension error, numerical error
    33 %: omega = \dot theta has been used, incorrectly
    25 %: correct answer deduced from incorrect reasons!
    25 %: correct starting formula, but no or little development from it
    10 %: some discussion is presented, but no clear direction is indicated
     0 %: answer missing, or completely incorrect

Part 2a (50 points) :

   100 %: practically perfect
    90 %: one sign error
    90 %: one numerical error
    80 %: one sign error, one numerical error
    80 %: two numerical errors
    75 %: missing the beta v_0 term
    75 %: dimension error, numerical error
    70 %: missing the beta v_0 term and one numerical error
    65 %: answer not in the perturbation form, numerical error
    65 %: dimension error, sign error, numerical error
    60 %: answer not in the perturbation form, missing the beta v_0 term
    55 %: zeroth order not good, dimension error, numerical error
    50 %: solved for a 2D case, did not obtain the final answer, zeroth-
          order good, first-order not good
    50 %: answer not in the perturbation form, dimension error, numerical
          error
    45 %: set up equation (with the  correct zeroth order) but did not
          evaluate the integral
    45 %: set up equation, partially evaluated the integral, zeroth order
          incorrect
    40 %: answer has time (t) in it, but zeroth order answer is ok
    33 %: answer has time (t) in it, and the zeroth order is incorrect
    25 %: correct starting formula, but no or little development from it
     0 %: answer missing, or completely incorrect

Part 2b (50 points) :

   100 %: practically perfect
    90 %: one numerical error
    90 %: one sign error
    80 %: two numerical errors
    80 %: one sign error, one numerical error
    70 %: dimension error, two numerical errors
    70 %: three numerical errors
    70 %: zeroth order missing (or incorrect), one sign error
    70 %: missing the beta v_0 term and one numerical error
    60 %: zeroth order missing (or incorrect), sign error, numerical error
    50 %: implicit equation for h set up; perturbation solution for h not
          given
    50 %: zeroth order correct, but did not extract the first order
          correction
    50 %: zeroth order is incorrect, dimensional error and numerical error
    45 %: set up equation (with the correct zeroth order) but did not
          evaluate the integral
    40 %: set up equation (with no or incorrect zeroth order) but did not
          evalute integral (or did it very incorrectly)
    33 %: solution incorrect in form (diverges as k goes to 0 and no zeroth
          order)
    33 %: answer has time (t) in it, the zeroth answer is incorrect
    33 %: potential energy incorrect, perturbation solution not obtained
    10 %: some discussion is presented, but no clear direction is indicated
     0 %: answer missing, or completely incorrect

Part 3a (34 points) : solution part 1: A cos (2 w t + phi_0)

   100 %: practically perfect
    95 %: left the solution as a complex function
    90 %: correct except for a (small) numerical error
    75 %: contains only one integration constant, not two
    70 %: contains three integration constant, not two
    50 %: no integration constants
    50 %: contains extra function, unnecessary and incorrect
    25 %: correct starting formula, but no or little development from it
     0 %: answer missing, or completely incorrect

Part 3b (33 points) : solution part 2: (B /(3 w^2)) cos (w t)

   100 %: practically perfect
    90 %: correct except for a (small) numerical error
    80 %: dimension error
    60 %: left in a general form with undetermined constant(s)
    50 %: sin(wt) instead of cos(wt) and incorrect factor
    25 %: correct starting formula, but no or little development from it
    10 %: some discussion is presented, but no clear direction is indicated
     0 %: answer missing, or completely incorrect

Part 3c (33 points) : solution part 3: - Bt / (4 w) cos (2 w t)

   100 %: practically perfect
    90 %: forgot to retain the t factor
    90 %: correct except for a (small) numerical error
    70 %: answer is proportional to t sin(2w t), not to t cos(2wt)
    50 %: answer is prop to sin(2wt) or cos(2wt), with no multiplicative
          t-linear term (or did not realize that the coefficient is
          infinity!)
    50 %: linear term and sinusoidal term are present, but did not extract
          a real function, or too many terms were obtained
    50 %: contains the correct function, up to a multiplicative factor, but
          with other unnecessary and incorrect functions and unnecessary
          integration constants
    40 %: sinusoidal function multiplied by a complicated function of t
    40 %: sinusoidal only (no multiplicative linear term) and contains
          general constant(s), undetermined
    25 %: correct starting formula, but no or little development from it
    10 %: some discussion is presented, but no clear direction is indicated
     0 %: answer missing, or completely incorrect

Part 4a (20 points) : U = 0.5 * (k1 x1^2 + k2 x2^2)

   100 %: practically perfect
    85 %: correct except that the sign is incorrect
    70 %: contains an unnecessary incorrect extra term
    50 %: only one term is correct
    33 %: 0.5 (k1 + k2) (x2 - x1)^2 or something similar
     0 %: answer missing, or completely incorrect

Part 4b (20 points) : k = 0.5 m (dot x1 + dot x2)^2

   100 %: practically perfect
    90 %: correct except for a (small) numerical error
    70 %: contains an unnecessary incorrect extra term, or two such terms
    33 %: got 0.5 m (dot x1^2 + dot x2^2) or similar
    25 %: got 0.5 m (dot x2^2 - dot x1^2)
     0 %: answer missing, or completely incorrect

Part 4c (20 points) : L  = K - U = 0.5 m (dot x1 + dot x2)^2 - 0.5 * (k1
                      x1^2 + k2 x2^2)

   100 %: practically perfect
    95 %: would have been perfect, were it not for (numerical) mistakes
          accumulated from previous part(s)
    85 %: correct except that the sign is incorrect
    40 %: very inconsistent with the answers for parts (a) and (b)
     0 %: answer missing, or completely incorrect

Part 4d (20 points) : m (ddot x1 + ddot x2) = - k1 x1 = - k2 x2

   100 %: practically perfect
    95 %: would have been perfect, were it not for (numerical) mistakes
          accumulated from previous part(s)
    85 %: correct, or logically correct from the previous part, except that
          sign or numerical factor is incorrect
    60 %: double dots missing somehow, (or some similar simple error) but
          OK otherwise
    40 %: very inconsistent with the answer of part (c)
    25 %: correct starting formula, but no or little development from it
    10 %: some discussion is presented, but no clear direction is indicated
     0 %: answer missing, or completely incorrect

Part 4e (20 points) : k1 x1 = k2 x2

   100 %: practically perfect
    95 %: would have been perfect, were it not for (numerical) mistakes
          accumulated from previous part(s)
    25 %: correct answer deduced from incorrect steps
    10 %: some discussion is presented, but no clear direction is indicated
     0 %: answer missing, or completely incorrect