= 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 }}}