Here you can see a histogram for final exam scores and read the full 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 }}}