Here you can see a histogram for final exam scores and read the grading rubrics used for the final exam.

Part 1a.1 (30 points) : eigen-freq = 0 and sqrt(3g/(2l))

   100 %: practically perfect
    80 %: a dimensional error, otherwise correct
    80 %: got 0, but the other one had a numerical error (e.g., due to
          mistake in building/solving matrix)
    60 %: got incorrect answers due to calculating omega^2 M incorrectly
    55 %: the M matrix is very incorrect (no coupling), leading to very
          trivial results (0 and sqrt(g/l))
    55 %: got the zero mode (0), and the other answer, incorrect, came out
          of nowhere
    50 %: the M matrix is very incorrect (no coupling), and there is an
          error in result
    50 %: did not get 0 (due to error in solving matrix), and the other
          answer has numerical error
    50 %: got only one (zero mode) by inspection or by matrix calculation
    50 %: did not get 0 (due to error in solving matrix), and the other
          answer has dimensional error
    33 %: K incorrect, and an unnecessary constraint was used
    25 %: K and U incorrect and answers are incorrect (no zero mode)
    25 %: correct (or close-to-correct) answer stated with no derivation
          (or very inconsistent derivation)
    25 %: correct starting formula, but no or little development from it
    25 %: only one degree of freedom (and so K is incorrect), and U is
          incorrect
    25 %: K and U are incorrect, and incorrect answers are written with no
          derivation
    10 %: some discussion is presented, but no clear direction is indicated
     0 %: answer missing, or completely incorrect

Part 1a.2 (30 points) : eigen-vectors: (1, 0) and (1, -3)  if q1 = x, q2 =
                        l \theta

   100 %: practically perfect
    90 %: got (1,1) instead of (1, 0), but good otherwise
    90 %: correct except for a (small) numerical error
    80 %: (1, -3) correct (or self-consistent), but got (0, 0) instead of
          (1, 0)
    80 %: got (1, -2) for the second one due to using l theta, not x + l
          theta, for m, using inspection
    70 %: (1, 1) instead of (1, 0), and the other has a numerical error
          (accumulated from a previous step)
    60 %: got (1, 0) correctly, but the other vector was not obtained
          (while the equation was set up) or came incorrectly out of
          nowhere
    50 %: the M matrix is very incorrect (no coupling), leading to very
          trivial results ((1,0) and (0, 1))
    50 %: made guesses and got (1, 1) and (1,-2)
    50 %: got incorrect answers due to calculating omega^2 M incorrectly
    50 %: sketches provided correctly for the two modes, but only
          qualitatively so
    33 %: got (0, 0) (or (1, 1)) instead of (1, 0) and the other answer is
          incorrect
    33 %: eigenvector equations were set up with an internal consistency,
          but neither solution is correct
    33 %: sketch provided for one mode only, and is only qualitative
    33 %: K incorrect, and an unnecessary constraint was used
    25 %: only one degree of freedom, and the potential is incorrect
    25 %: correct starting formula, but no or little development from it
    25 %: correct (or close-to-correct) answer stated with no derivation,
          or very inconsistent derivation
    25 %: used a 3x3 formalism, incorrectly, and matrix with error, and
          gave solution with error
    25 %: K and U are incorrect, and incorrect answers are written with no
          derivation
    10 %: some discussion is presented, but no clear direction is indicated
     0 %: answer missing, or completely incorrect

Part 1b.1 (20 points) : hit center of mass, l/3 from the top

   100 %: practically perfect
    80 %: "the center of mass of the pendulum system" is stated, without
          giving "where along the rod"
    60 %: the center of mass for the final state is considered, not the
          initial state
    33 %: torque consideration was done, but incorrectly
    25 %: correct answer deduced from incorrect reasons!
    10 %: some discussion is presented, but no clear direction is indicated
     0 %: answer missing, or completely incorrect

Part 1b.2 (20 points) : vf = v0 / 4, using momentum conservation

   100 %: practically perfect
    75 %: v0 / 3 was given, since the mass of the mud was not included
    75 %: v0 / 5 was given, since the mass of the mud was included twice
    60 %: mass factor error, but the momentum conservation concept was used
    50 %: answer given, but no rationale given
    10 %: some discussion is presented, but no clear direction is indicated
     0 %: answer missing, or completely incorrect (e.g., used the kinetic
          energy conservation)

Part 2 (100 points) : v2 = 2v and E2 = 4K/3 > 0: hyperbola (never coming
                      back)

   100 %: practically perfect
    90 %: correct except for a (small) numerical error
    85 %: v2 = 2v part is OK, but E2 > 0 is stated without any reasoning
    75 %: v2 incorrect (after using the momentum conservation alone,
          correctly), but the sign of E2 correct
    65 %: v2 had a minor error, which led to a completely wrong conclusion
    65 %: v2 = 2v part is OK, but the energy discussion is incorrect
    50 %: v2 = 2v part is OK (with a possible minor numerical mistake), but
          the energy discussion is basically absent
    50 %: v2 incorrect (and the kinetic energy conservation was used,
          incorrectly), the sign of E2 turned out to be correct
    50 %: v2 was not obtained, and the kinetic energy conservation during
          collision, which is invalid, was used to show E2 > 0
    25 %: v2 incorrect (used elastic collision, incorrectly), the sign of
          E2 incorrect
    25 %: correct answer stated with no proper derivation at all
    25 %: correct starting formula, but no or little development from it
    25 %: v2 incorrect, and the energy discussion is incomplete
    25 %: v2 incorrect (used kinetic energy conservation, instead of
          momentum conservation), the sign of E2 incorrect
    25 %: v2 incorrect, and the sign of E2 incorrect; starting formulas
          have error
    10 %: some discussion is presented, but no clear direction is indicated
     0 %: answer missing, or completely incorrect

Part 3a (30 points) : gvec = -2 AG \hat{\rho} / rho (out), - 2 A G rho
                      \hat{\rho} / R^2 (in)

   100 %: practically perfect
    90 %: correct except for a (small) numerical error
    80 %: correct except a dimensional error
    50 %: only one of the two is OK
    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 3b (30 points) : Phi = 2 AG log (rho/R) (out), AG (rho^2 - R^2) / R^2
                      (in)

   100 %: practically perfect
    95 %: would have been perfect, were it not for (numerical) mistakes
          accumulated from previous part(s)
    95 %: everything correct, except a dimension error in the integration
          constant
    90 %: correct except for a (small) numerical error
    85 %: correct except that the sign is incorrect
    75 %: correct, except that Phi is not continuous at R
    50 %: only one of the two is OK
    25 %: correct starting formula, but no or little development from it
    25 %: correct answer deduced from incorrect reasons!
     0 %: answer missing, or completely incorrect

Part 3c (30 points) : SHM with omega^2 = 2 AG / R^2

   100 %: practically perfect
    95 %: would have been perfect, were it not for (numerical) mistakes
          accumulated from previous part(s)
    50 %: after an overly complicated calculation, which went wrong, made a
          guess that it is a SHM
    33 %: could only say only "some kind of oscillatory motion" despite the
          correct potential
    33 %: could say some kind of oscillation, based only on qualitative
          discussion
    33 %: SHM based on qualitative discussion only
    25 %: correct answer deduced from incorrect reasons!
    25 %: correct answer stated with no derivation
    10 %: some discussion is presented, but no clear direction is indicated
     0 %: answer missing, or completely incorrect

Part 3d (30 points) : time = 0.5 T_{SHM} = sqrt( pi^2 R^2 / (2 AG) )

   100 %: practically perfect
    90 %: correct except for a (small) numerical error
    80 %: correct except a dimensional error
    50 %: nearly correct expression but time depends on a dynamical
          variable (such as theta or x)
    50 %: stated period / 2, without being able to calculate it
    25 %: correct starting formula, but no or little development from it
    25 %: stated period, without being able to calculate it, or presented
          without any derivation
    10 %: some discussion is presented, but no clear direction is indicated
     0 %: answer missing, or completely incorrect

Part 3e (30 points) : normal force = m omega^2 d

   100 %: practically perfect
    90 %: normal force = Fg cos theta, not expressed in terms of given
          parameters such as d
    80 %: correct except a dimensional error
    50 %: the constraint set up correctly (y), L in error, answer in error
    33 %: said "normal force" without actually finding it, or finding it
          very incorrectly
    33 %: the constraint set up incorrectly, and the Lagrange equation for
          y did not appear
    10 %: Hooke's law force is stated with no use of the Lagrange
          multiplier formalism or equivalent
    10 %: just said "constraint force"
     0 %: answer missing, or completely incorrect

Part 4a (30 points) : L = 0.5 m R^2 (t1d^2 + t2d^2 + w^2 (t1^2 + t2^2) -
                      w_k^2 (t2-t1)^2 - w_g^2 (t1+t2)^2) where t1 = theta1,
                      t2 = theta2, t1d = dot(theta1), t2d = dot (theta2),
                      w_k = sqrt(k/m), w_g = sqrt(g/R)

   100 %: practically perfect
    90 %: correct except a sign error
    85 %: one part (e.g., the w^2 part) has error
    80 %: kinetic energy has serious errors
    80 %: all correct except that one of the two potentials (due to k and
          g) is missing
    80 %: all correct except that the w^2 terms are missing
    70 %: two parts (e.g., the w^2 part and the gravity part) have errors
    70 %: kinetic energy has serious errors, as well as one other part
    65 %: kinetic energy has serious errors and one of the two potentials
          (k or g) is missing
    60 %: kinetic energy has serious errors as well as some errors (sign,
          dimension) in two other parts
    40 %: all terms have errors
     0 %: answer missing, or completely incorrect

Part 4b (30 points) : L_M = 0.5 (2m) R^2 ( dot(Q)^2 + (w^2 - w_g^2) Q^2 ),
                      L_int = 0.5 (0.5 m) R^2 ( dot(q)^2 + (w^2 - 2w_k^2 -
                      w_g^2) q^2 )

   100 %: practically perfect
    95 %: would have been perfect, were it not for (numerical) mistakes
          accumulated from previous part(s)
    90 %: correct except for a (small) numerical error
    85 %: by mistake, the gravity terms resulted in a linear term
    85 %: correct except a dimensional error
    80 %: correct except that the w^2 part is not retained
    80 %: correct except that the gravitation potential is not retained (to
          2nd order)
    60 %: the w^2 term is not retained and somehow Q --> dot(Q) and q -->
          dot(q) in some parts
    40 %: separation of variables were forced, but with error (due to
          sinusoidal function not expanded)
    40 %: separation was forced and one of the two (L_int, e.g.) is grossly
          incorrect
    35 %: some calculation, but the separation of variables is not
          demonstrated at all
    25 %: correct answer stated with no derivation
    25 %: correct starting formula, but no or little development from it
     0 %: answer missing, or completely incorrect

Part 4c (30 points) : H_M = 0.5 (2m) R^2 ( dot(Q)^2 + (w_g^2 - w^2) Q^2 ),
                      H_int = 0.5 (0.5 m) R^2 ( dot(q)^2 + (2w_k^2 + w_g^2
                      - w^2) q^2 ), H = H_M + H_int,  all (H, H_M, H_int)
                      are conserved

   100 %: practically perfect
    95 %: would have been perfect, were it not for (numerical) mistakes
          accumulated from previous part(s)
    75 %: all correct (or self-consistent), except for the lack of remark
          on conservation
    70 %: H's have serious sign errors but the conservation property is OK
    50 %: only one is correct
    33 %: H_M and H_int were not obtained, or were obtained incorrectly,
          but the correct statements regarding their conservation were
          given
    25 %: H_M and H_int were not obtained correctly, and the statements
          regarding their conservation is with an error
    25 %: correct starting formula, but no or little development from it
     0 %: answer missing, or completely incorrect

Part 4d (30 points) : P_Q = (2m) R^2 dot(Q), p_q = (0.5m) R^2 dot(q); none
                      of these are conserved

   100 %: practically perfect
    95 %: would have been perfect, were it not for (numerical) mistakes
          accumulated from previous part(s)
    80 %: correct non-conservation statements, but expressions have
          dimensional errors
    50 %: P_Q and p_q are (nearly) correct, but did not say whether they
          are conserved or not
    50 %: only one argument is correct
    50 %: correct statements on non-conservation, but (practically) no
          formulas given for the momenta
    50 %: P_Q and p_q are (nearly) correct, but said that both P_Q and p_q
          are conserved
    33 %: P_Q and p_q are quite incorrect (or not given), and said that
          both of them are conserved
    25 %: P_Q and p_q are quite incorrect, and did not say whether P_Q and
          p_q are conserved
    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 (30 points) : U_{eff,M} = 0.5 (2m) R^2  (w_g^2 - w^2) Q^2,
                      U_{eff,int} = 0.5 (0.5 m) R^2 (2w_k^2 + w_g^2  - w^2)
                      q^2

   100 %: practically perfect
    95 %: would have been perfect, were it not for (numerical) mistakes
          accumulated from previous part(s)
    50 %: only one argument is correct
    33 %: qualitative argument only
    25 %: correct starting formula, but no or little development from it
     0 %: answer missing, or completely incorrect