Statistics
Here is the histogram for the final score distribution. It is an OK distribution in terms of the shape. The average is rather low, though. This is the reason why I curved the final scores, adding 45 points to the raw score (shown here) to get the re-normalized score, which was then used for computing letter grades.
Grading Rubrics
Here is the complete grading rubrics used for this final. For the problems that you attempted you have gotten one of the percentile grades per each part.
Part 1a (20 points) : Hyperbola since E_{2m,2v} = 2(4T + U) = 4T > 0, where T and U are those for the mass m.
100 %: practically perfect
90 %: correct except for a (small) numerical error
85 %: all math correct, while the answer is not (parabola, instead of hyperbola)
60 %: T and U arguments used, but incorrectly.
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 1b (30 points) : -3v for m and 0 for 2m from the p conservation and the relative velocity flip.
100 %: practically perfect
75 %: all math correct, except for a fatal physics mistake that the non-collision solution was selected and the collision solution was thrown out as unphysical!
60 %: equations set up correctly, but incorrect answers came out
50 %: one of the two equations set up incorrectly, so got incorrect answers
40 %: error in starting equations (coeff restitution sign error, e.g.), and then inconsistent/incorrect answers
33 %: incorrect result due to applying the general result of the CM frame to the LAB frame
25 %: only one equation set up, and no or incorrect answer given
25 %: correct starting formulae, but no or little development from them
10 %: some discussion is presented, but no clear direction is indicated
0 %: answer missing, or completely incorrect
Part 1c (10 points) : Hyperbola for m since E_m = 9T + U = 7T > 0 (where T and U are for its initial state) and 2m in a line fall to M since it has no velocity/angular-momentum.
100 %: practically perfect
90 %: fully logical conclusion derived carefully from (b), but physically incorrect
80 %: answer correct except saying "parabola or hyperbola" for m2
80 %: answers correct, but reason for hyperbola not explained
50 %: got only one of two correct
50 %: while unphysical and incorrect, answers are simple direct consequences of incorrect answers of (a,b)
25 %: while unphysical and incorrect, one answer is a simple direct consequence of incorrect answers of (a,b)
25 %: correct starting formula, but no or little development from it
15 %: one answer is correct, but with no proper derivation
10 %: some discussion is presented, but no clear direction is indicated
0 %: answer missing, or completely incorrect
Part 2a (15 points) : g = -2GA rho / R^2 (inside), -2GA/rho (outside)
100 %: practically perfect
90 %: correct except for a (small) numerical error
85 %: correct except that the sign is incorrect
75 %: correct except for the sign error and some (simple) numerical errors
75 %: Gauss law set up correctly, but incorrect solution for one case (inside or outside)
60 %: Gauss law set up correctly, but incorrect answers for both cases (inside and outside)
50 %: Gauss law set up correctly, but no answers derived from it
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 2b (15 points) : Phi = GA rho^2/R^2 (inside), 2GA ln(rho/R) + GA (outside); other choice of constants possible
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
85 %: all ok, except that Phi is not made continuous at R
70 %: sign error and Phi not continuous at R
50 %: one of the integrals done incorrectly, starting from (a).
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 2c-i (8 points) : A SHM (with k = 2GAm/R^2)
100 %: practically perfect
33 %: the goal was clearly stated, but could not be derived
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 2c-ii (7 points) : time = tau/2 = pi / omega = pi R / sqrt(2GA)
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
75 %: all fine, but a critical mistake of confusing omega^2 and omega
75 %: the full period, tau, is given
50 %: ans = tau/2 = pi / omega is stated with no final answer
33 %: ans = tau = 2 pi / omega (incorrect) is stated with no final answer
10 %: some discussion is presented, but no clear direction is indicated
0 %: answer missing, or completely incorrect
Part 2d (15 points) : U_eff = 0.5 (k - m omega_0^2) rho^2 from H = \dot{rho} p_rho - L
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
33 %: used H = T + U, incorrectly, and got the correct answer due to a previous mistake
33 %: simply used H = T + U, incorrectly, and therefore got an incorrect U_eff
33 %: treated theta as a dynamical variable and got an incorrect answer
10 %: some discussion is presented, but no clear direction is indicated
0 %: answer missing, or completely incorrect
Part 2e (10 points) : omega_c = sqrt(k/m) = sqrt(2GA) / R
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
75 %: sign of d^2 U_eff/d rho^2 mis-interpreted
70 %: the second derivative evaluated correctly, and then completely misinterpreted
50 %: correct qualitative discussion with no/incorrect omega0
50 %: the second derivative evaluated incorrectly, leading to an incorrect answer
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 2f (10 points) : lambda(t) = 2m rho \dot{rho} omega_0, normal force = lambda(t)/rho = 2m\dot{rho}omega_0 (balances the Coriolis force)
100 %: practically perfect
75 %: correct set up, correct formal EOM, but error in calculating lambda(t)
50 %: correct qualitative discussion with no quantitative result
25 %: correct starting formula, but no or little development from it
20 %: L + constraint formalism set up (and not solved), but for r, not for theta
10 %: some discussion is presented, but no clear direction is indicated
0 %: answer missing, or completely incorrect
Part 3a1 (5 points) : T for hoop = M \dot{x}^2
100 %: practically perfect
90 %: correct except for a (small) numerical error
80 %: incorrect by an additional unnecessary term
75 %: incorrect due to misunderstanding of the relationship between x and theta
75 %: translation + rot, but not simplified using the constraint
60 %: accounted for the CM motion only
40 %: accounted for rotational motion only
0 %: answer missing, or completely incorrect
Part 3a2 (10 points) : T for rod = 0.5 m (\dot{x}^2 + 2l \dot{x}\dot{theta} + (4 / 3) l^2 \dot{theta}^2)
100 %: practically perfect
90 %: correct except for a (small) numerical error
75 %: translation (of CM) + rotation, but incorrect for the translation part
60 %: accounted for the CM motion only
50 %: translation + rotation, but incorrect for both terms
40 %: only rotational term (around CM) is included
33 %: incorrectly calculated and mixed symbols with the hoop
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 3a3 (5 points) : U = mgl theta^2 / 2
100 %: practically perfect
90 %: correct except for a (small) numerical error
85 %: correct except that the sign is incorrect
75 %: prop to theta^2, but some non-trivial error in the coeffcient (dimension mismatch)
75 %: got const * cos theta, and then approximated it to a const (!) by doign a small angle expansion
70 %: left as const * cos theta, without doing small angle expansion
65 %: error due to using sin instead of cos
60 %: left as mgy
10 %: some discussion is presented, but no clear direction is indicated
0 %: answer missing, or completely incorrect
Part 3a4 (5 points) : M matrix = [ [2M+m m] [m 4m/3] ] if q1 = x and q2 = l theta
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
50 %: a 3x3 matrix, due to not using the rolling w/o slipping constraint
50 %: incorrect, since only diagonal terms are kept (despite non-diagonal terms in T)
33 %: incorrect in both diagonal and off-diagonal terms, and not symmetric
10 %: some discussion is presented, but no clear direction is indicated
0 %: answer missing, or completely incorrect
Part 3a5 (5 points) : A matrix = [ [ 0 0 ] [ 0 k ] ] where k = mg/l
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 %: correct except that the sign is incorrect
75 %: the position of k is incorrect
50 %: incorrect due to extra term(s)
50 %: a 3x3 matrix, due to not using the rolling w/o slipping constraint
33 %: theta (or other generalized coordinate) in the A matrix, quite incorrectly
10 %: some discussion is presented, but no clear direction is indicated
0 %: answer missing, or completely incorrect
Part 3a6 (10 points) : omega = 0 and sqrt (3 (m + 2 M) / (m + 8M)) * sqrt (g/l)
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
75 %: math correct, but did not get 0 freq mode, and did not realize that flaw in physics
75 %: 0 and a finite frequency, the latter logically right (being derived from a very simple, incorrect, model) but physically incorrect
75 %: 0 and a finite frequency, the latter incorrectly derived
75 %: calc went wrong, but recognized the qualitative modes that should result
70 %: incorrect secular equation, and 0 and finite freq from it
50 %: correct starting formula, no answer given after some work
50 %: math incorrect, did not get 0 freq mode due to math error
33 %: incorrect secular equation, and incorrect answers (no 0 freq)
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 3a7 (5 points) : T = [ 1 0 ] for omega = 0, [ -m 2M+m ] for the other mode
100 %: practically perfect
95 %: would have been perfect, were it not for (numerical) mistakes accumulated from previous part(s)
75 %: T incorrectly identified from omega = 0, from physical reasoning
75 %: the 2nd answer absent and qualitative discussion given for it
75 %: the 2nd answer is tried but no result on that after some computations
60 %: got only one correct
50 %: qualitatively correct diagrams given, but those only
50 %: both have errors, while the approach was correct
40 %: only rough qualitative descriptions given
33 %: incorrect math, leading to incorrect eigenvectors for both
33 %: answers given as symmetric and anti-symmetric pairs (incorrect in this case as the set up is not symmetric)
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 3b1 (10 points) : initial conditions applied properly (using the T^{-1} matrix)
100 %: practically perfect
95 %: would have been perfect, were it not for (numerical) mistakes accumulated from previous part(s)
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 3b2 (5 points) : x = (1 - cos (omega_2 t)) m l theta_0 / (m + 2 M) and theta = theta0 cos (omega2 t)
95 %: would have been perfect, were it not for (numerical) mistakes accumulated from previous part(s)
50 %: general formulae given, with constants not determined
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 4 (50 points) :
50 %: made correct guesses about the normal mode shapes (at least three of them) based on symmetry
50 %: incorrect U, but ended up getting correct eigenvectors
33 %: the shapes of two normal modes guessed correctly
25 %: some discussion is presented (with somewhat incorrect U, e.g.), but no clear direction is indicated (as to how to deal with the 4x4 problem) with no symmetry argument noted
25 %: correct starting formula, but no or little development from it
25 %: incorrect A, and incorrect solutions obtained trivially from it
10 %: only T is given
0 %: answer missing, or completely incorrect