HW & Grading
HW rules :
HW due Sept 3, 2009
TA: Sally Jones
HWs and Quizzes:
Final HW :)
1) Draw a truss with at least 3 bars, more if you dare. Load it with forces as you like. Draw this neatly. Calculate the tensions in all of the bars.
2) Assume a motor following the standard motor law always operating at its peak (i.e., like an electric motor at fixed
voltage, not peak power). The transmission is lossless gear box. The motor has peak power P_p. Assume the wheels never skid. Other key numbers:
m = total mass of car (1000 kg)
P_p = 100 hp
rho = density of air (1 kg/m^3)
A = cross sectional area (2 m^2)
v = present speed of car (variable)
c_d = drag coefficient (1.0)
F_f = constan friction force (1000 N)
F_0 = force at wheel if the car is still (depends on gear ratio).
v_f = speed of car when motor supplies no torque.
The car is slowed by air drag:
Air drag = rho * A * c_d * v ^2/2.
and by a constant friction force F_f.
a) Assuming the gearing is such that the stall force at the wheels is F_0 = 10,000 N (this would be the
gearing that would make the car almost skid if the friction coefficient was 1). Graph the position vs time for 20 seconds.
b) Find the fixed gear (characterized by F_0) that minimizes the time to reach 60 mph. What is that time?
c) Assume that you had a continuously variable transmission so that the P_p was always delivered to the wheels. What then would be the time to reach 60 mph? (This problem is easier than (b)).
Here are solutions to HW-10
This homework is all about solving ODEs. Here's a simple example.
[t zarray] = ode23(@myrhs,[0 1], 1);
function zdot =myrhs(t,z)
zdot = z;
You can look at samples: odedemo1 and odedemo3
You can use these to get started. But, by the time you are done with this homework you should be able to walk up to Matlab and do problems like the homework without looking at any samples.
1) Notice that the integral of sin(t) from 0 to pi is also the solution of this ODE:
dx/dt = sin(t) with x(0) = 0evaluated at x(t=pi). See how accurate this is using ode23 or ode45.
2) Assume your car has a mass of 1 kg. Pick any gear ratio you want. Assume its peak power is 10 watts. Assume the transmission is perfect. Assume a fixed gear ratio G. Assume the motor follows the classic straight line motor law. Neglect air friction. Pick any G you want.
a) Find the time to go 15 m from a standing start.3) Solve this system of equations for ANY functions f,g,h that you want and ANY initial condition:
b) Make this time as short as you can by finding the best possible G.
dx/dt = f(x,y,z)Plot x vs y. See if you can find functions f,g,h so that the plot looks interesting (you decide what is interesting).
dy/dt = g(x,y,z)
dz/dt = h(x,y,z)
Here are solutions to HW-9
0. If you didn't succeed with the homework last week, do that now. You can use the lecture notes (or video) from Oct 29, and you can look at the code posted, but you should do your own work.That means, for example, that if your dog ate your homework before you handed it in, and you couldn't talk with anyone else, that you would be able to do the homework again on your own with no notes.
1. For all of this week keep these assumptions: 150 lbm rider, P_p = .5 hp, omega_f = 180 rpm, no air friction and no transmission losses. For this problem the gear ratio G is the ratio of the angular velocity of the crank to the velocity of the bike. This is also the reciprocal of the distance traveled for each radian of crank turning.
a. Plot the peak speed as a function of G (easy).
b. Plot the time to reach 20 mph as a function of G (more than a quickie).
c. Find x at 10 s. Do not use the calculus formulas for v(t) and x(t). Instead use numerical integration. In informal computer language, that's
x(0) = 0, v(0) = (0), t = 0
calculate acceleration from vend loop
t_new = t + h
x_new = x + h*v
v_new = v + acceleration * h
Compare your numerical integration answer to that from the calculus formulas.
Make them agree! (You need this skill for the next problem.)
2. Now include air friction. The bike is slowed by air friction according to the formula:
F_drag = c_d * rho * A * v^2/2
rho is the density of air, use 1.2 kg/m^3
A is the frontal area of the bike and rider, use 1 m^2
c_d is an empirical drag constant, use 0.9.
This says its harder to go through a dense fluid than a light one (e.g., water than air), that big things push more fluid around than
small things, and that when you double the speed the drag goes up by a factor of 4. The drag coefficient c_d depends on shape; for trout and airplanes it less than 0.9.
Note, you cannot assume that the power of the pedaling goes just into kinetic energy of the bike. It goes into air friction as well, with the air dissipation being v* F_d.
a. For a given gear ratio there is a given peak speed of the bike (the peak speed is the speed that is reached when there is no acceleration). If G is too big, then the speed is limited by the rider not being able to pedal fast enough. If the G is too small (called a "high gear") than there's not enough force to move the bike fast through the air. Make a plot of the peak speed of the bike as a function of gear ratio G.
b. For a gear ratio of G = 1 m^-1, plot the velocity as a function of time for starting from a stationary start. Plot for 60 seconds (You will probably have to calculate this with numericalintegration).
Here are solutions to HW-8
0. Reading assignment (30 minutes).
Read the section on units in Ruina and Pratap.You can find the book online by googling, say, Ruina textbook. Or click here: http://ruina.tam.cornell.edu/
1. Units check. (Unless you are having fun, spend no more than 15 minutes on this.)
a. Repeat the constant power bicycle calculation done in lecture. Not the calculus, just the final putting in of numbers. Complete the calculation with numbers, all the way to a numerical answer in seconds, using the method of2. Another bike acceleration problem (Unless you are having too much fun, stop working after 4 hours).
"multiplying by 1".
b. Do the calculation above again, but with a different unit system as follows. From each of the identities below you
also get a 1, for example:
5280 ft = 1 mile => 1 = (5280 ft) / (1 mile)c. Compare your answers from (a) and (b) above. If the agreement is less than one part in 10^3, find conversion factors for both the SI biased (a) and English biased (b) calculations to get them to agree better. Can you get 6 digit agreement?
1 hores power = 550 ft lbf / s
1 lbf = 1 lbm * g
g = 32 ft / s^2
We will continue with this problem in the coming weeks. So you should understand the reasoning involved well.
SetupAssume that for the duration of a short race, that the torque (moment= M) a bicycle rider can apply to the bicycle cranks obeys
M = M_0 - c * omega -- (A)where:
M_0 = "M sub zero" = the "stall torque"
omega = the angular velocity of the crank
c = a constant showing that the rider can apply less torque when pedaling harder
omega_f = M_0/c = the fastest the rider can turn the cranks (at no load).
[Aside: this is the standard model used for electric motors. On the Omega-M plane it is a straight line between the stall torque and the free spinning speed. It is approximately what you discovered in lab last week. For muscles the curve most commonly used here is not a straight line but a hyperbola. Look up "Hill's model" of muscle contraction if your are curious about it. We're just using this straight-line electric-motor model of a muscle here for simplicity.]
The propulsive force F from this torque is dependent on the gear ratio, following the energy equation for transmissions (assuming no transmission losses).
Power in = Power outThus the relevant measure of the bike transmission ratio is the gear ratio, G = omega/v and the thrust is
M * omega = F * v
F = G * M = (omega/v) * M -- (B)remember that M is going to depend on omega (see eqn A).
a. Go look at a real bicycle. Use a ruler to measure things if you need to. Make a sketch. And, for some gear combination of the bike, find G.
b. Assume that, with appropriate gearing, the peak power of the bicycle rider is 0.5 hp and the fastest he/she can turn the cranks is omega_f = 180 rpm. Find M_0 and c.
c. Assuming the numbers above, a bike and rider total mass of 150 lbm:
i. plot the speed vs time for starting from a stopThis calculation (c) can be done using formulas you derive or using numerical integration. In either case your work needs to be well documented and clear and your plots should be done with Matlab.
ii. plot the position vs time
iii. how long does it take to reach 20 mph?
iv. what distance has been covered when 20 mph is reached.
1. Tune up your quiz score. How? Do it again well and hand in with homework. (Its in the link here). Do all 3 problems. It will be scored as 2*(TOTAL of 3)/3. Your recorded quiz score for course grading will be the average of this re-do and your in-class quiz score. Higher standards here than in class. Follow the standard homework academic integrity policies (not the quiz or exam policies) posted on the course www page.
a. Use words to explain all your algebra. Your answers should be convincing to someone who does not know how to do the problem.
b. For the integration problem do it 3 ways, always showing the results on the computer and commenting on the results.
i. Using loops to do a sum
ii. Using the SUM command to avoid loops
iii. Using a fancy accurate method such as QUAD, QUADL, QUADGK, SYM/INT or ERF, and making the answer as accurate as you can.
iv. For method (ii) do the integral several times for the widest possible range of 'n'. Make a "convergence plot" with x axis being log(1/n) (log base 10) y axis being the log of the difference between your answer using method (ii) and the more exact answer from method (iii). Explain the features of this plot as best you can.
3. Assume a bicycle rider (mass = m) starts from a stop and has constant power P. What are the velocity and position as functions of time? Make plots using m = 150 lbm and P=.5 horse power and for a enough time to show up to speeds of 20 mph. How long does it take to reach 20 mph?
a. Be careful with units. Explain your unit conversions carefully.
b. This calculation is NOT a realistic one because
i. at the start it involves infinite forces and infinite accelerations.
iii. it neglects air friction.
But we will fix it up next week.
Matlab only. Continue with tutorial lessons. Hand in a note saying how much time you spent on them and what lesson you have completed competently. If you have completed all lessons hand in something that shows off your skills.
1. Keep working on Matlab lessons as time allows. Report your progress and amount of effort. We have tentatively scheduled an extra help session for Sat Oct 3 in the afternoon. This past weekend didn't work out (family issues).
2. A sphere of radius R is cut off by a plane that is a distance H from the center. What is the volume of the truncated sphere.
Calculate this by adding up discs that are parallel to thecut plane (as opposed to the cylinders used in lecture). Your result is V(H,R), (volume V as a function of H and R).
3a. Write a matlab function called 'myroot' or something like that so that
finds a real root of the function
A x^3 + B x^2 + C x + D = 0
with at least 3 digit accuracy. For example,
ans = 1.0000 or ans = 2.0000 or ans = 3.0000.
[Why? Because x^3 - 6 x^2 + 11 x - 6 = (x-1)(x-2(x-3). You can use these commands from the Matlab symbolic toolbox to check this (or pencil and paper will work fine too): syms, expand, factor]
You just need to find one root. If you want to be fancy and find all real roots, if there are more than one, then that is ok. Just not needed. Think McGyver. Don't try to be fancy, at least to start with. Pretend instead that you just want the answer and you have to figure it out, using what you know, one way or another. Don't look in books, work it out yourself or with a friend.
Here is one approach if you don't think of another. Its called "bisection" or informally "divide and conquer".
A) Assume that the root is bigger than -100.Feel free to make plots and list intermediate calculations and so on.
B) Plug in x = -100 and see if the result is positive or negative.
C) Plug in 0 and see if the result changed signs. If so you know there is a root between -100 and 0.
Plug in -50. From the sign you can tell if the root is between -100 and -50 or between -50 and 0.
Then plug in -75 or -25, appropriately. And so on.
D) If the polynomial has the same sign at 0 as at -100 then plug in +100 and do the same song and dance.
E) Code this all up with a termination rule for when you have the root located with an accuracy that
you like. IT should all be in loops with IF statements and so on so it happens automatically.
3b. As a challenge, see how many different schemes you can cook up that serve the purpose, both using fancy Matlab commands (like ROOTS and FZERO, FSOLVE) or using other methods that you code up yourself.
1. The usual Matlab assignment.
a) Spend at least 2 hours doing Matlab tutorials. If you have completed the tutorial lessons, then go on to learn the Matlab command QUAD using Matlab help. Compare the numerical integrals with analytic ones for some problems of your choice.
b) Report, in a sentence or two, how much time you spent learning matlab and how far along you are in the tutorial lessons.
2. Do the attached tolerancing Homework from Professor Voelker. Your whole solution should be one page at most. Note that you are not alone if you have trouble seeing, from the drawing, how the brake actually works. Which parts move, and in which directions, when the brake is in operation? Do your best to guess about this when answering the questions.
Solution to Voelker Problem
1. Continue with tutorials for another 2 hours. Write
"I worked ______ hours on the tutorials this week and feel I have mastered through lesson/tutorial _______."
No showing off this week, at least not to hand in.
2. Assume constant density. Assume that you know that
Pressure in a fluid is = rho*g*z (or y, whichever one you take as up) + C (some constant).
Or you could say its rho*g*h, where h is the distance from the zero-pressure surface. Repeat carefully, in a way that you find fully convincing, the lecture calculation/derivation that
Bouyant force = rho* g * (displaced volume).
In the lecture some steps were done a bit quickly [like where there was (cos theta)/(cos theta)]. Do those steps more slowly and with enough sketches so that you find the argument totally convincing (to you, pretending you are skeptical). Vector calculus not allowed. OK, afterwards you can do it one more time with vector calculus if you have those skills.
3. Assume a uniform constant density fluid with density rho_m (that's rho with a subscript m, for, say, milk).
A ball with radius R and uniform density half as big
rho_b = rho_m /2
floats in it. It floats half submerged. This is all in a big pot. Next a lighter liquid that does not mix, say oil, is poured on top and it submerges the sphere. This second liquid has half the density of the ball, rho_o (o for oil) is
rho_o = rho_b/2 = rho_m /4.
Exactly how far up or down does the ball go (from its orginal position) when submerged by the pressure from the oil above?
a) Try to explain in convincing words why the ball goes up or down.
b) Do the quantitative calculation (If there is complicated calculation, do it as best and accurately as you can).
(In lecture on Thursday Sept 24 it was demonstrated how to do this homework problem using numerical integration. Here is the program shown in lecture: sphere.m )
1. Spend one hour on Matlab Lessons (eventually you want to more-or-less master all the lessons).
a) How far did you get?2. Spend an hour showing off with Matlab. This could be a continuation or improvement of what you did
b) For at least one problem you have try to get help on it from at least 3 different kinds of Matlab "help"
(See pratap Text intro for a list of help options).
What did you seek help on?
What source was the best?
last week. It need not be a graph.
3. The Facts of Life
a) Consider this experiment.
i) Take two more or less equal glasses of cold tap water.What do you think? Psychological what-does-the-teacher-think arguments are not legit! Are the effects you can think of large or small effects? Should the difference in melting times be small or large?
ii) Put about two Tablespoons =~ 1.5 centiliters of salt in one and stir it around.
(3 teaspoons to a tablespoon. 2 tablespoons to an ounce. One ounce is about 29 grams(!). One gram of water has a volume of 1 milliliter, etc.)
iii) Put two ~equal ice cubes in the two glasses, one in each glass.
iv) Don't stir.
v) Which one melts faster?
b) Do the experiment.
c) Explain the result. If the effect was a small one then your explanation should be about small differences. And vice versa.
d) Design an experiment that illuminates something about your result.
e) Write this all up. If you use books or www sources (not recommended, the more you have to confront your lack of solution, the more you gain) cite them and exactly what you learned from which source.
For this, and future work, you can write things by hand, and annotate printed things
by hand. Typing is ok, also. Just not required.
1. Write at the start of HW: "I have read, understood, and agree to follow the posted lab and homework guidelines and academic integrity rules for this class."
2. The matlab tutorials, are posted under the tab 'course documents' on Blackboard.
Spend 3 hours sitting at a computer and working through the tutorials posted, you are not expected to complete them all in one go. Please answer these questions.
a) How much time did you actually spend? (e.g., "I spent 4.5 hours at the computer doing the tutorials.")
b) Which tutorials did you complete? (Again, a readable sentence.)
c) What did you have trouble with? (Again, and henceforth, your answer should make sense to someone who has not read the question you are answering.)
3. Spend 1 hour doing a calculation or plot of some kind that you think is interesting or fun or shows off what you have learned. Hand in something that shows what you have done. This should include, formatted nicely:
a) a description
b) the matlab commands and
c) the output.