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Lesson 1
Read through section 2-6.
Be prepared to work problems next Monday.
Units – make sure you can convert units.
 I don’t want any mistakes on units, that is NOT the point of this class.
o Can you do unit algebra? Yes, do it. No then do the following.
 Learn to do it BEFORE you get very far along – come to office
 Convert all units into:
 Feet, Pounds, Slugs, Seconds, do the problem then convert
to the desired unit for the answer.
 Meter, Newton, kilogram, seconds, do the problem then
convert to the desired unit
Real world versus the classroom
 Objective is to give enough real world stuff that you can actually use the stuff but
do enough classroom junk so you can graduate
 Tests will always be over classroom junk, I’ll tell you when its real world stuff.
Real stuff can often help you understand but I’ll not ask you to do it on a test.
Approximation
 Things are way too complex to include everything so we approximate.
 Trick is to throw out as much as you can but not too much. Comes with
experience.
 Swing a pendulum with and without Earth rotation.
Solving equations – How to count unknowns and equations. (Start with linear equations.)

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4x 3y
3
3y
2
4
y
1
9
x
5
4
,y
4x 3y
1
3
6
,
6 4x
3
or
There are an infinite number of solutions.
What if there are more equations than unknowns?
o If they are consistent 4 x 3 y 6, 8 x 3 y 9, 16 x 6 y 18 you can
y

3y
Can you solve this
,
How many unknowns? How many equations?
What if there are more unknowns than equations?
x

6, 8 x
1
,x
5
4
find an answer
o If they are inconsistent 4 x 3 y 6, 8 x 3 y 9, 16 x
no answer.
Can you always find a solution when variables equal equations?
o Solve
x
3
4x 3y
6, 8 x
3
2 y
4
6y
12
,
y
2
4x
3
or
6y
10
there is
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

These are dependent and consistent, there is an infinite number of
solutions.
o Solve 4 x 3 y 6, 8 x 6 y 11 there is no solution. These are
inconsistent.
In dynamics sometimes variables cancel out, Can you solve this?
4xz 3yz


6 z, 8 x z
6yz
x
20 z
2
2, y
3
,
You can typically spot
this because of a common term. If its ugly often you can “expect” certain ones to
drop out, mass, coefficients (friction), Normal forces, etc.
You will receive partial credit IF you list the “unknown variables” and number
the equations. In fact you get more credit for counting and convincing me that you
can solve them than you do for actually solving them.
Create a table, chart or graphic in your notes that summarizes this material. Pass it
to your neighbor and have them check it.
Solving differential equations – We integrate them in this class.






1
x t
3t 4x t
0, x 0
1
3t 4x
0, x 0
1, x 0
Solve
,
You need one condition for every integration.
t
2
8
8 3 t2
x t
1
8
8 16 t
t3
Solve
,
Solve a=x, dV/dt = x, dV/dt dx/dx = dV/dx dx/dt = x = dV/dx V = x, V dV = x dx,
½ v^2 = ½ x^2 + c
In the real world, you actually solve differential equations.
How many “initial conditions” do you need?
Position
 Form a triad, one of you will be throwing a party at your house, the second one
needs directions to your party, the third is an observer identifying what kinds of
things you have to say when you give your directions. Be ready to report out.
 Origin, directions, distance
 Origin is assumed known by all parties. It is arbitrary BUT your directions are
consistent with its choice.
 Directions, assumed known by all parties. It is arbitrary BUT your ...
 Distance ...
 Is there a difference between distance traveled and position? What is it?
Velocity is the time derivative of position.
 Can you have a zero position now but a non-zero velocity? Vice versa?
 Plot position versus time, slope is speed
 Area under velocity curve is position + constant
Acceleration is the time derivative of velocity.
 Same questions as velocity
 What is acceleration and deceleration?
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Handy Dandy Formulas for Constant A
 V=vo + a(t-to)
 X=xo + vo (t-to) + ½ a (t-to)^2
 V^2=vo^2 + 2 a (x-xo)
 Memorize these!
Degree of freedom
 How many motions can it have?
Using the chain rule to take derivatives
 What does it mean that a variable is a function of time?
 For each of the functions below determine the first and second derivatives of x
with respect to t.
4
t Do it by substitution and by the chain rule.
 First let
o x Cos
^2 Sin
o x
^2 Sin
o x
x
L
Sin
o
3 t^ 2
o x L^2 Cos
 Repeat the above if the function of is not known
 Repeat the above if L is a function of time.
Vectors
 We use unit vectors to talk about directions. We use standard vectors but you can
(should) use as many as convenient. I and j. Convert (quickly) all the following
into i and j (they are all unit vectors).
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All non-unit vectors are a number times a unit vector. 3 i +4 j
o Find magnitude, make it a unit.
o What is the angle between it and the i? the j?
Dot product is mag first mag second Cosine angle between, also show the math
formula
o Find angles of 3i + 4 j using dot product
o Find angles of -3i + 4j draw a picture and use dot product
o Do again using 3i-4j
o In 2D using the dot product is like killing a mosquito with a cannon but it
also works in 3D where drawing pictures is a pain
Cross product is mag first mag second times Sine of angle between, points in
direction perpendicular to both using right hand rule
o As homework find a way to compute cross products of two dimensional
vectors (i,j) and 3D vectors (i,j,k)
When you differentiate a vector for now do the following.
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o Write the vector as a distance times I or j
o Take the derivative of all the distances, leave the i and j alone.
o T^2 i + 4 t j, derivative is 2t i + 4 j