15 mathematical fundamentals need working knowledge of algebra and basic trigonometry if you don’t...
Post on 19-Dec-2015
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1
Mathematical Fundamentals
• Need working knowledge of algebra and basic trigonometry
• if you don’t have this then you must see me immediately!
2
Algebra Review
• Exponents - Square Roots
52
exponent
5 * 5 = 25
23 = 2 * 2 * 2 = 8
25 = 251/2
= 5
3
Order of Operations
• Solve the following problem
(12 + * 3)2-1 + 3 * 2 - (8/4) - 52 - 6/2 = ???23
4
5
Order of Operations
• (1) parentheses, brackets, and braces
• (2) exponents, square roots
• (3) multiplication and division
• (4) addition and subtraction
6
Order of Operations ProblemSOLUTION
(12 + * 3)2-1 + 3 * 2 - (8/4) - 52 - 6/2 = ???2
3
1. parentheses(12+ *3)
1a. * 3 = 2
1b. 12 + 2 = 14
1c. 8/4 = 2
2
3
2
3
7
Order of Operations ProblemSOLUTION
(12 + * 3)2-1 + 3 * 2 - (8/4) - 52 - 6/2 = ???2
3
2. exponents52 = 25
3. multiplication & division(12 + *3)2 = 14*2 = 28NOTE: 14 was calculated in steps 1a and 1b.
6/2 = 3 3*2 = 6
2
3
8
Order of Operations ProblemSOLUTION
(12 + * 3)2-1 + 3 * 2 - (8/4) - 52 - 6/2 = ???2
3
Substitute into equation
28 -1 + 6 - 2 - 25 - 3 = 3
9
Trigonometry
• field of mathematics focusing on relationships between sides of and the angles within a right triangle
10
Trigonometry Review
ca
b
a = “opposite” sideb = “adjacent” sidec = “hypotenuse” = angle
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SOHCAHTOA
ca
b
4 Basic Relationships1. a2 + b2 = c2
(Pythagorean Theorem)
2. sin = opp/hyp = a/c
3. cos = adj/hyp = b/c
4. tan = opp/adj = a/b
a = “vertical component”b = “horizontal component”c = “resultant”
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Two types of TRIG problems
Type A
Type B
Given Solve For c & a & b
a & b c &
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TYPE A Problem
v = 10 m
/s
40o
b
aGiven:
c = 10 m/s = 40 degrees
Find: a and b
sin 40o
=a
10 m/s
10 m/s * sin = a
10 m/s10 m/s40
o
cos 40o
=b
10 m/s
10 m/s * cos = b
10 m/s10 m/s40
o
b = 10 m/s * cos 40 = 7.66 m/sa = 10 m/s * sin 40 = 6.43 m/so o
14
Type B Problem
100 lb
400 lbc
Given: a = 400 lb, b = 100 lbFind: c and
a2 + b2 = c2
(400 lb)2 + (100 lb)2 = c2
160000 lb2 + 10000 lb2 = c2
170000 lb2 = c2
c = 412.3 lb
atan = b
400 lbtan= 100 lb
tan= 4
tan-1 (tan ) = tan-1(4)
= 76.0o
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Inverse Trig Functions
If sin is a trig function
then sin-1 is aninverse trig function
:inverse trig functions simply “undo” trig functions
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SOHCAHTOA
• SOH Sine = Opposite/Hypotenuse
• CAH Cosine = Adjacent/Hypotenuse
• TOA Tangent = Opposite/Adjacent
17
25
20o
b
a
Calculate the vertical (a) and horizontalsides of this right triangle.
18
25
20o
b
a
sin 20 = a
25
ocos 20 =
b
25
o
a = 25 (sin 20)a = 8.55
b = 25 (cos 20)b = 23.49
19
10
15c
Solve for the lengthof the hypotenuse (c)and the angle, .
20
10
15c
c = 152 + 102
c = 325
c = 18.03
tan = 15
10
= tan-1 (1.5)
= 56.3
o
21
UNITS
• Use the SI system– AKA Metric System– 4 basic units
• length -- meter• mass -- kilogram• time -- second• temperature -- degree Kelvin
(Celsius)
Radio Flyer
50 lbs
45o
Billy pulls on his new wagon with 50 lbs of force at an angle of 45 .How much of this resultant force is actually working to pull the wagon horizontally?
Vector Resolution Example
Fx
Fy
F
F = Fx + Fy
45o
F = magnitude of F = 50 lbs
cos 45 =
sin 45 =
Fx
Fy
F
F
o
o
Fx
Fy
F
F = Fx + Fy
45o
= cos 45
= sin 45Fy
Fx F
F
o
o
Fx
Fy
= 50 lbs (cos 45 ) = 50 lbs * 0.707 = 35.4 lbs
= 50 lbs (sin 45 ) = 50 lbs * 0.707 = 35.4 lbs
o
o
Radio Flyer
50 lbs
45o
Sometimes the magnitude of a force iswritten more simply as
Fx = 35.4 lbsFy = 35.4 lbs
Only the force acting in the x-direction acts to move the wagon forward
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Vector Decompositionaka Vector Resolution
Any vector can be expressed as a pair of two component vectors
these vectors 1) must be perpendicular to each other
2) are usually horizontal and vertical
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Given the polar notation of a vector, decompose it into vertical and horizontal components (Cartesian coordinates).
•sx = |S|cos = 10(.766) = 7.66 m• sy = |S|sin = 10(.643) = 6.43 m
= 40o
S = 10
m
10cos(40)
10sin(40)
Vector Decomposition
x
y
Vector Composition(aka Vector Addition)
• to add 2 vectors must consider both magnitude and direction
• the sum of 2 or more vectors is known as a resultant vector
• if the vectors have the same direction then you may add the magnitudes directly
+ =
• vectors are in opposite direction– resultant vector points in direction of longer
vector– size of resultant vector is the difference
between the component vectors
+ =
• vectors are pointed in different, non-parallel, direction
• graphical solution - TIP-TO-TAIL method
+
• TIP-TO-TAIL method
place the tail of the 2nd vector at the tip of the 1st vector
connect the tail of the 1st vector to the 2nd vector
+ =resultant
vector
Resultant vector isthe diagonal of theresulting parallelogram
• TIP-TO-TAIL method is the preferred method when adding more than 2 vectors– include more vectors by attaching their tail
to the open tip in the diagram
+ + + +
+ + + +
Vector Example
Graphically compute the resultant force acting on the femoral head.
Two Forces Acting on the Hip
muscle
bodyweightW
resultant forceacting on the femoral head
Fm
W
W
R
R = Fm+W
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• Vectors can be added by placing the tail of each vector at the tip of the previous one.
• The sum of all of these vectors is called the resultant vector. It connects the tail of the first vector to the head of the last vector.
resu
ltant
Vector Addition
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• Finding the horizontal and vertical components of each vector makes it easy to find the resultant.
Vector Addition
38
• Simply add all of the vertical lines for the vertical component and add all of the horizontal lines for the horizontal component. Be sure to pay attention to the sign of each of the lines.
resu
ltant
Vector Addition
39
• Use the following formulas to convert the coordinates into polar notation:
•
• = arctan
|S|
Vector Addition
2y
2x sss
Sx
Sy
40
y
x
S2 = 3m, 165o
S1 = 6m, 40o
41
y
x
S2 = 3m, 165o
S1 = 6m, 40o
42
Sx1 = |S1|cos1 = 6(.799) = 4.60 m
Sy1 = |S1|sin1 = 6(.643) = 3.86 m
Sx2 = |S2|cos2 = 3(-.966) = -2.90 m
Sy2 = |S2|sin2 = 3(.259) = .78 m
Sx = 4.60 - 2.90 = 1.70 m
Sy = 3.86 + .78 = 4.64 m
y
x
S2 = 3m, 165o
S1 = 6m, 40o
43
• |S| = = 4.94 m
• = arctan = 69.9o
Polar Notation