= Statistics = Here is the histogram for the midterm score distribution. It is a decent distribution. However, I do worry about those students below 110 points or so (according to this plot, there are 23 students below 110) -- please do come and see me if you belong in that group. On the other hand, the exam clearly showed a pretty solid general grasp of core subjects by the majority of students, and I have seen some particularly very nice solutions -- Kudos to you, who did very well! <> = Grading Rubrics = Here is the complete grading rubrics used for this midterm. For the problems that you attempted you have gotten one of the percentile grades per each part. {{{ Part 1.p1 (15 points) : Work done by gravity = -mgh 100 %: practically perfect 85 %: correct except that the sign is incorrect 25 %: correct starting formula, but no or little development from it 0 %: answer missing, or completely incorrect Part 1.p2 (15 points) : Work done by hand: m(g+a)h/2 + m(g-a)h/2 = mgh 100 %: practically perfect 40 %: Used a, for both paths! (correct ones to use: g+a and g-a) 25 %: correct starting formula, but no or little development from it 25 %: correct answer stated with no derivation 0 %: answer missing, or completely incorrect Part 1.p3 (10 points) : The net work = -mgh + mgh = 0, confirming the W-E theorem. 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 0 %: answer missing, or completely incorrect Part 2.p1 (20 points) : First conservation principle. 100 %: practically perfect 67 %: Invariance of L, missing or incorrectly stated 67 %: the conserved quantity missing or mis-identified (e.g. said L is conserved) 33 %: conserved quantity OK, but the experimental reproducibility and the L invariance not (or incorrectly) discusssed 33 %: experimental reproducibility OK, but L invariance and conservation are missing or incorrect Part 2.p2 (20 points) : Second conservation principle. 100 %: practically perfect 67 %: invariance of L, missing or incorrectly stated 67 %: the conserved quantity missing or mis-identified (e.g. said L is conserved) 33 %: experimental reproducibility OK, but L invariance and conservation are missing or incorrect 0 %: answer missing, or completely incorrect Part 3.p1 (10 points) : EOM: m ddot{x} + k x= mu_k m g 100 %: practically perfect (or equivalently in the energy view) 85 %: correct except that the sign is incorrect (or equivalently in the energy view) 85 %: correct except the spring force incorrectly written as k (x+A) 33 %: incorrect, using mu_k mg dot{x} 0 %: answer missing, or completely incorrect Part 3.p2 (10 points) : General solutions to the EOM 100 %: practically perfect (or equivalently in the energy view) 95 %: would have been perfect, were it not for (numerical) mistakes accumulated from previous part(s) 75 %: x_p good, but x_c is missing an integration constant 70 %: x_p good, but x_c is not (exponential instead of oscillatory) 50 %: x_p not correct (missed some factors) and x_c missing an integration constant 50 %: x_c OK, but x_p incorrect or missing 40 %: x_c OK except missing an integ constant, and x_p not OK 33 %: x_c incorrectly obtained, and x_p missing 33 %: incorrect EOM, and inconsistent solutions for them 33 %: incorrectly applied the characterstic equation 33 %: x_c OK except missing an integ const, and x_p missing 0 %: answer missing, or completely incorrect Part 3.p3 (10 points) : Correct determination of two constants 100 %: practically perfect (or equivalently in the energy view) 95 %: would have been perfect, were it not for (numerical) mistakes accumulated from previous part(s) 95 %: followed the correct procedure, but symbol A got confused 75 %: followed the correct procedure, but symbol A confused and did not take the real part of x 50 %: x_0 satisfied, but not dot{x_0} 50 %: dot(x_0) satisfied but not x_0 40 %: some development but with no end results 25 %: correct starting formula, but no or little development from it 0 %: answer missing, or completely incorrect Part 3.p4 (10 points) : Identification of the turning point as dot{x} = 0 100 %: practically perfect (or equivalently in the energy view) 33 %: identified as 1/4 period or 3/4 period (incorrect) 0 %: answer missing, or completely incorrect Part 3.p5 (10 points) : Turning point value: x = -A + 2 mu_k mg / k 100 %: practically perfect 95 %: would have been perfect, were it not for (numerical) mistakes accumulated from previous part(s) 50 %: time identified correctly, but forgot (?) to give the x value 50 %: the answer has correct ingredients, but it is incorrect and steps are unclear 0 %: answer missing, or completely incorrect Part 4.p1 (10 points) : work-energy theorem: change in KE = net work 100 %: practically perfect 85 %: correct except that the sign is incorrect 0 %: answer missing, or completely incorrect Part 4.p2 (10 points) : net work = work done by friction = -km \int dt v^4 100 %: practically perfect 85 %: correct except that the sign is incorrect 50 %: incorrect power of speed 25 %: did -k m \int dt v^2 instead 0 %: answer missing, or completely incorrect Part 4.p3 (10 points) : the 0th order solution for v_x(t), x_y(t) used for the work-by-friction term 100 %: practically perfect 75 %: correct except thata = omega t was used (not applicable here) 50 %: v_y correct, but v_x missing or incorrect 50 %: v_x correct but v_y is not 0 %: answer missing, or completely incorrect Part 4.p4 (10 points) : alpha = kv_0^3 / g (up to other multiplicative constants) 100 %: practically perfect 95 %: would have been perfect, were it not for (numerical) mistakes accumulated from previous part(s) 50 %: close but not quite right (wrong powers of v_0, e.g.) 0 %: answer missing, or completely incorrect Part 4.p5 (10 points) : v_f = v_0 (1 - C alpha) 100 %: practically perfect 90 %: correct except for a (small) numerical error 40 %: correct form but incorrect result 33 %: missing numerical factor(s), leading to inconsistent result, and the final form is not v_0 (1-C alpha) 25 %: incorrect form (no zero-th order term), incorrect result 25 %: some computation, but result it not evaluated clearly 0 %: answer missing, or completely incorrect Part 5 (50 points) : T = 2 pi sqrt(l/g) for a simple pendulum 100 %: practically perfect 95 %: everything good, except I = ml^2 is not used 90 %: everything good, except a sign error in the EOM 90 %: everything good, except T is not obtained (only w0), or is incorrect. 80 %: everything good (logically at least), except some error in the EOM (dropped l or something similar) 70 %: everything good, except a major flaw in EOM (extra dot{theta}, e.g.) 60 %: EOM correct, but solution and T are not 60 %: slightly incorrect EOM (sign error) and incorrect solution from it, while T = 2pi/w0 is noted 50 %: incorrect EOM, consistent sols of it, but no T (only w0) 50 %: incorrect EOM , no/incorrect solution, but T = 2pi/w0 noted 40 %: slightly incorrect EOM (sign error) and no/little/incorrect development from it 35 %: incorrect use of energy conservation, which led to a correct answer due to missteps 33 %: incorrect EOM and inconsistent/non-existent solution from it 33 %: incorrect use of energy conservation, which led to an incorrect answer 25 %: answer is stated, but no/little derivation to back it up 25 %: correct starting formula, but no or little development from it 0 %: answer missing, or completely incorrect Part 6.a (15 points) : L = 0.5(m+M)\dot{r}^2 + 0.5 mr^2\dot{theta}^2 - Mgr 100 %: practically perfect 85 %: all OK, except two many generalized coords (3 or more) 85 %: all OK, except a sign mistake in T 80 %: all OK, except that the kinetic energy of M is missing 70 %: all OK, except that the kinetic energy of M is missing and too many generalized coords (3 or more) 70 %: U nearly correct (sign mistake), T nearly correct (# of gen. coord. incorrect), and L = T - U is correct 70 %: U incorrect, but ok otherwise 70 %: T incorrect, but ok otherwise 60 %: T is nearly correct (# of gen. coord. incorrect), U incorrect, and L = T - U is correct 60 %: U is nearly correct (sign mistake) but T is incorrect, and L = T - U is correct 33 %: T and U both incorrect but L = T - U is correct 0 %: answer missing, or completely incorrect Part 6.b (10 points) : Yes. No friction, air resistance, etc. Or, partial L/ partial t = 0 and H = T + U = E. 100 %: practically perfect 70 %: said "because there is no friction or any other outside force" (gravity is an outside force) 60 %: said "because there is nothing doing work on the system" (gravity does work) 50 %: correct general answer but lacks the discussion of non-conservative forces or the explicit t dep of L 33 %: correct answer (conserved), but no reason given 0 %: answer missing, or completely incorrect Part 6.c (15 points) : p_theta = m r^2 \dot{theta} is conserved 100 %: practically perfect 75 %: p_theta is incorrect slightly (incorrect power of theta, e.g.), ok otherwise 50 %: said that all of p_theta, p_r (or p_z) are conserved. 50 %: found p_theta, but did not realize/state that it is conserved (or said \dot{theta} is conserved, incorrectly) 50 %: stated that p_theta is conserved, but it is not/incorrectly found 0 %: answer missing, or completely incorrect Part 6.d (10 points) : circular motion with r = sqrt[3]{p_theta^2 / (m M g)} 100 %: practically perfect 75 %: ok, except that r is not expressed in terms of p_theta 75 %: basic set up is good, but minor incorrect step(s) leading to an incorrect answer 50 %: slightly flawed equation, and did not express the answer in terms of p_theta 50 %: basic set up OK, but did not use \dot{r} = 0, and got no good answer 25 %: correct starting formula, but no or little development from it 0 %: answer missing, or completely incorrect Part 6.e (10 points) : Yes, try to push or pull and let go -- can prove that it comes back by restoring force! 100 %: practically perfect 75 %: correct answer and partially valid argument 25 %: correct answer stated with no/incorrect derivation 0 %: answer missing, or completely incorrect }}}