Study advice:

***Do problems and get help with this goal in mind. be able to do assigned homework problems

from beginning to end,

with no help from book, notes, solutions, people, etc.,

by yourself, without looking up even one teeny tiny its bitsy small thing.
And be able to convincingly explain every step. ***

You can get to that state by working things out by yourself or by using all manner of help, including friends, TAs, Google, YouTube, old solutions you find online, etc. Always keep the final goal, above, in mind. Your goal is not a piece of paper with a correct solution. Your goal is to gain the ability described above. You don't know if you have this ability unless you test it; if you did use help, then afterwards start the problem over by yourself (without looking up even one infinitessimal thing).

Again: you can't know if you can do a problem without help unless you have actually done it without help. More or less, anyone who achieves such competency with the assigned problems will get an A. Want an A+ level of understanding: to the extent that time, energy, ability and ambition allow you, see if you can do other problems that are similar to the assigned problems (without help, even the first time).

Rather than getting through more and more difficult problems with help, try to increase the level of difficulty of problems you can do without help. This prepares you better for exams and builds your useful skills. Just getting through homework, with help, has some (maybe small) value; it builds your work ethic and concentration skills. But mastering some of the homework, gives bigger and more long-lasting skills.

Homework Policy and Guidelines:
These homework guidelines are our best guess as to how to help you with this. These guidelines also make dealing with homework manageable for us.

OLD HOMEWORK SOLUTIONS

Use these at your own risk. They might not be correct. And even if correct, copying them might not be the most efficient way to learn.

Assigned Problems

Problems are from RP (Ruina and Pratap) unless otherwise specified or written out.

Readings

If problems are assigned from a given section, reading that section is implicitly assigned. Additionally, you should occasionally glance at the front and back tables, and, through the semester, learn what is there. Read other material from the book as needed and as described in HW 1.

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HW 1

Due Saturday, Feb 13, 2021 at 11:59 PM

Problems:

1) Spring and mass hung from ceiling. Find ODE and solution.
2) Find solution with Euler's method, and compare
3) Front-wheel drive ar driving up hill at constant speed, find needed friction coefficient $\mu$.

Homework assignment 1 in detail.

Reading guide for this week and coming weeks.
Read and understand these sections as soon as possible, but definitely before the first prelim. Most of it is review.
(If you know that you have mastered a given topic, obviously you don't have to read about it again.)

- Skim the Detailed table of contents so that you know what is in the book (pg 3, pdf page = book page + cover pages = 3+4 = 7)
- Skim the front and back tables, so you know where to find that reference material
- "To the student" starting on marked page 20 (add 4 to get the pdf page, 24 in this case)
- Chapter 0, What is mechanics?
- Chapter 1: Vectors (adding and 3 kinds of multiplication)
- Chapter 2: Free-Body Diagrams (master section 2.3)
- Chapter 3: ODEs
- Chapter 4: Units
- Chapter 5-1- 5.3: 2D statics of one object
- Chapter 7: Transmissions and Mechanisms
- Chapter 10: Dynamics in 1D

Especially related to HW 1:

- Figure 2.2 from the text about equivalent force at bearing
- Section 2.4 on friction forces in Free Body Diagrams
- Sections 3.1 and 3.2 on Differential Equations
- Section 5.2, Equilibrium of one object (e.g., "three force bodies")
- Section 5.3 on friction in statics problems
- Section 7.1 on springs
- Section 10.3: Mass and spring, harmonic oscillator

HW 1 SOLUTIONS

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HW 2

Due Saturday, Feb 20, 2021 at 11:59 PM

Problems:

4) Write a Matlab function that can solve (all?) ODEs from Math 2930. Use the following functions and variables.

Write a Matlab function myEulerSolver which you would call like this:

[tarray zarray]  = myEulerSolver(@myrhs, tarray, z0, p);

And the first line of the function is this:

function  [tarray, zarray]   = myEulerSolver(eqn, tarray, z0, p)

To use this myEulerSolver you would have to another function myrhs that contains your ODE. Its first line is this.

function zdot = myrhs(t,z,p)

The words used above have these meanings:

t	time, a number representing the present instant of interest
`tarray`	a column vector of times. The first element is t_0, the last is t_end. The difference between two successive entries is the time step.
`z`	a column vector of state variables at time `t. For example z = [x;v] in 1D particle dynamics.`
`zarray`	Each row is `z'` (transpose of `z`) at the corresponding time `t` of `tarray`. The first row is `z0'.`
`z0`	Initial conditions. The values of state variables at t=0. For example `z0 = [x0;v0] in 1D particle dynamics.`
`p`	A struct of parameters needed for the ODE. For example `p` could have in it `p.m, p.k, p.c`, etc.
`myrhs`	The ODE you want to solve. The last line must assign values to a column vector `zdot.` Inside my EulerSolver the variable `eqn` will evaluate as the function handle @myrhs.
`zdot`	A column vector of the rates of change of the state variables at the given time.
`myEulerSolver`	Your program that can numerically solve any ODE from math 2930. The inputs and outputs are described above.
`function`	A basic matlab command used as the first word in a block of code that defines a Matlab function.

4b) Test your function above with the harmonic oscillator and with exponential growth

5) text problem 10.1.26 Quadratic drag on a bullet. Think of it as jello or water or mud if you don't like the given application.
(numerical solution desired, compare with an analytical solution if you have time).
6) text problem10.2.3 This is easy, just vocabulary practice.
7) 10.2.11 A simple energy problem using some weird archery words.

This week's homework has no separate pdf file.

HW 2 SOLUTIONS

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HW 3

Due Saturday, Feb 27, 2021 at 11:59 PM

8) Write and test your own ODE2 using the midpoint method.

[tarray zarray] = myMidpointSolver(eqn, tspan, z0,options);

where

options.maxtime is the amount of time at which, if not complete, the computer should give up, and
options.RelTol is a measure of the relative error.
tspan should be two numbers, a start time and an end time.

If implementing options.RelTol is too much programming complexity for you, replace it with options.nsteps which is the number of intervals. Using an ODE of your own choice, make a plot of the time it takes to get a given accuracy, vs accuracy, for your new solver and for your old solver. You should find that when you want accurate solutions, your new solver is much faster. You only need a few points on this curve. But make one of those points be associated with an accuracy that takes your ODE1 solver on the order of a minute or more. You should find that you gain little by using ODE2 if the accuracy needed is not high. But if high accuracy is needed, ODE2 should save you lots of time.

9) 10.2.10 The fall distance is the wall height + the leg bending. A simple problem.

10) 10.2.16 Constant power acceleration. This is a bit subtle and takes a bit of thought.

11) 10.3.8 Trampoline. Part c requires careful thought because the period of time of contact with the trampoline is not half the period of the associated harmonic oscillator (because of gravity, the feet don't leave the trampoline at the mid-point of the oscillations). This is not a quick problem.

12) 11.1.9 Basic damped oscillator problem. A numerical solution is perfectly fine. So is an analytic one. Your choice. Best to do both and compare.

13) 10.4.14 Two masses, three springs and a dashpot. Write and solve the equations.

HW 3 SOLUTIONS

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HW 4

Due Sunday, March 7, 2021 at 11:59 PM
14) 11.3.3 This is a very simple conceptual question, basically asking the definition of normal mode.
15) 11.3.8 A simple problem intended to make you think about motions of mulit-DOF systems
16) 11.3.7 For part (a) Use intuition or guessing, or iterative guessing, or eigenvalues and eigenvectors on Matlab, or any combination of these.
Add in part (c): For the same initial conditions in part b, plot x1, x2, & x3 vs t (all on the same plot, for 0<= t <= 20s.
17) Redo any problems you didn't do well at, or want to improve or expand on. Label these clearly. No need to hand in the old version.

HW 4 SOLUTIONS

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HW 5

Due Saturday, March 13 Sunday, March 14, 2021 at 11:59 PM
If you are short of time, I recommend doing the problems in this order:
18, 23, 22, 21, 19, 20

18) 10.5.6 A simple problem taking you through the concepts and vocabulary of 1D collisions.
19) 10.5.10 Tests if you can keep your hat on while calculating a sequence of collisions. And the answer is interesting.
20) 10.5.12 A bit of collision theory (the relation between e and energy dissipation)
21) 12.1.22 In spirit, this is extremely close to a 3D particle statics problem.
22) 12.1.26 Add part d) What is the general motion (what is the full set of trajectory shapes that is possible)? This problem is genuinely interesting. It has all the look of an intractable non-linear problem but turns out to be a simple linear problem.
23) 12.2.19 A very simple problem to show if you know what the words mean.

HW 5 SOLUTIONS

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HW 6

Due Saturday March 20 at 11:59 PM:

24) 12.1.31 This problem should expand your understanding of parabolic-flight ballistics to the more realistic ballistics of things where air drag is important. Add part (e): Assume, in some consistent units, that $m = 1, g = 10$ and $c = 1$ in some consistent unit system. Assume the initial launch speed is 1000. Forget the wall. Find the best angle to achieve the maximum range. What is that angle and range?

25) 12.3.5 A computer simulation of a missile trajectory
26) 13.1.10 A cute simulation of 3 balls in space: Mongomery's 8.
27) 13.2.7 2 balls colliding in 2D.
28) 13.2.10 Note that 13.2.7 is not of the standard form, so you can use this program to check your answer to 12.2.7, but not to generate it

HW 6 SOLUTIONS

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HW 7

Sat March 27 at 11:59 PM

35) 14.1.6 Easy pulley problem
36) 14.1.14b Slightly more involved pulley problem
37) 14.1.26 Pulley with spring.
38) 14.2.11 Simple constrained-object problem
39) 14.2.14 Il-posed constrained-object problem, why?
40) 14.2.25 Car braking. Long statement is basically just a sequence of hints for a problem that could be stated briefly.

HW 7 SOLUTIONS

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HW 8

Due Sunday, April 4

41) 15.1.1 Basically a vocabulary lesson/test about circular motion
42) 15.1.15 A simple test of whether you can work with the circular motion ideas
43) 15.2.21 Everything (or most things) you should know about a simple pendulum
44) 15.2.30 Another circular motion problem, bead on a hoop with friction
45) 15.2.34 A classic energy/circular motion problem,

HW 8 SOLUTIONS

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HW 9

Due Sunday April 11 at 11:59 PM
The problems in [brackets] are optional. Do them if you have the time (you haven't yet spent 6 hours, or it pleases you to spend more than 6 hours).

46) 16.1.8 Computer graphics, using rotations to repeatedly draw a rotated drawing.
[47) 16.2.14 Pendulum with unknown hinge location. A simple problem. But you have to think to turn the words into sensible equations]
48) 16.2.22 Very simple gear problem
[49) 16.2.24 A more challenging problem, with a math and computer flavor, about angular velocity. Could take an hour or so.]
50) 16.4.10 A bent 'L' spins about a hinge at a constant rate. A quick easy mechanics problem
51) 16.4.20 2 Gears. An easy mechanics problem (almost just kinematics)
52) 16.4.34 Multipart pendulum problem. For parts (a,b) answer in terms of sensible variables. A computer will help with some of the plots. This problem will take at least a good hour to do well.

HW 9 SOLUTIONS

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HW 10

Due Sunday April 18, 2021 at 11:59 PM

Full credit for this HW if at the top you write "I worked n hours on this HW" and n>=3. 80% if n>=2. 60% if n>=1.
The homework will not be graded for content and there will be no peer grading.

53 ) 17.1.1 Simple kinematics problem. Nothing hard.
54) 17.1.12 Javelin. Somewhat involved kinematics. (If you are short of time, skip this one.)
55) 17.2.7 Block in space with a force. Computer code and solution. Some work, but not difficult.
56) 17.2.9 Suspended mass with cut springs. Simple instantaneous dynamics problem.

HW 10 SOLUTIONS

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HW 11

Due Wed April 28 at 11:59 PM

57) 17.3.3 Plotting things about the motion of a point on a rolling tire. A nice computer plot is best
58) 17.4.6 Spool is pulled by a rope. A real problem, but not super hard.
59) 17.4.9 Napkin ring. This problem requires careful setup and real thought.
60) 17.4.23 Disk in cylinder. A real problem. Takes time and care.
61) 17.5.8 Acrobat. Nothing too hard. But you have to keep your hat on through the various steps.
62) 18.1.5 Simple polar coordinates problem, at least once you understand polar coordinates.

HW 11 SOLUTIONS

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HW 12

Due Sunday May 2, 2021 at 11:59 PM
(this is, Andy thinks, a relatively easier HW assignment)

63) 18.1.6 Like 18.1.5, but concerning acceleration. May take some thought. But not hard once you get it.
64) 18.1.10 Very simple vocabulary test, but with a neat drawing.
65) 18.2.5 Very simple problem once you understand rotating coordinates.
66) 18.3.2 Bug walks on line on rotating turntable. (parts (b) and (c) are optional).
67) 19.1.12 Bead on rotating stick. Not easy, not too hard.

HW 12 SOLUTIONS

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HW 13

Due Sunday May 9 at 11:59 PM

68) 18.4.1 Slider crank. Involved kinematics problem. Have to keep your hat on.
69) 18.4.4 Interacting rods. Hint: vel of C = vel of C.
70) 18.4.10 Interacting rods, considering acceleration. Hint: velocity of E = velocity of E; and acceleration of E = acceleration of E.
71) 19.1.1(a-h): You should neglect gravity. Easy: a vocabulary test.
72) Double pendulum. No gravity. 2D. First link has length LaTeX: L_1 and is spun by a motor at a given (known) non-constant rate $LaTeX: \omega(t)$ . Second link has mass LaTeX: m and moment of inertia LaTeX: I^G about its center of mass G which is a distance LaTeX: L_2 from from the end of the first rod. Find the equations of motion (find $LaTeX: \ddot \phi$ , the angle between the extension of the first rod and the second rod). This problem generalizes the lecture 2 ways: 1) The rotation rate of the platform (disk or first link) is not constant; and 2) The second link is a rigid object and not just a point mass.

HW 13 SOLUTIONS

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HW 14

NOT due.
Graded based on hours spent only. No peer grading. No content grading.
At top write "I worked ___ hours on this HW."
n = 6+ => 100%; 4-6 => 80%; 2-4 => 70%; 1-2 => 50%; 0-1 => 0%.

74) 19.3.10 Mass in slot. 2 DOF. Part f is a challenge (think of it as extra credit).

75) 19.3.27 Shaken inverted pendulum. Animate the solution. To get stability, use very fast, small-amplitude shaking. See lecture for physical demonstration.

76) 19.2.16 Which way does a bike accelerate? Needs very careful thought and set up.

HW 14 SOLUTIONS

** ALL HW Below is not assigned *
These problems were assigned in one previous year or another.

77) 19.3.1 Particle on a springy leash. Test of concepts. Not hard.
78) 19.3.3 The same problem as above, in disguise.
79) 19.3.8 Yo yo. Pretty easy too.

80) 19.3.10 Mass in slot. 2 DOF. Part f is a challenge.
81) 19.3.16 Pendulum on a cart. Easy problem.

82) 19.3.28 Double pendulum. A special, relatively easy, case of a hard problem.
83) 19.3.30 Rimless wheel. Long involved problem.