== 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 }}}