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