Mathematical Fundamentals

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• Need working knowledge of algebra and basic trigonometry

• if you don’t have this then you must see me immediately!

Algebra Review

• Exponents - Square Roots

exponent

5 * 5 = 25

23 = 2 * 2 * 2 = 8

25 = 251/2

Order of Operations

• Solve the following problem

(12 + * 3)2-1 + 3 * 2 - (8/4) - 52 - 6/2 = ???23

Order of Operations

• (1) parentheses, brackets, and braces

• (2) exponents, square roots

• (3) multiplication and division

• (4) addition and subtraction

Order of Operations ProblemSOLUTION

(12 + * 3)2-1 + 3 * 2 - (8/4) - 52 - 6/2 = ???2

1. parentheses(12+ *3)

1a. * 3 = 2

1b. 12 + 2 = 14

1c. 8/4 = 2

(12 + * 3)2-1 + 3 * 2 - (8/4) - 52 - 6/2 = ???2

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

(12 + * 3)2-1 + 3 * 2 - (8/4) - 52 - 6/2 = ???2

Substitute into equation

28 -1 + 6 - 2 - 25 - 3 = 3

Trigonometry

• field of mathematics focusing on relationships between sides of and the angles within a right triangle

Trigonometry Review

a = “opposite” sideb = “adjacent” sidec = “hypotenuse” = angle

SOHCAHTOA

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”

Two types of TRIG problems

Type A

Type B

Given Solve For c & a & b

a & b c &

TYPE A Problem

v = 10 m

aGiven:

c = 10 m/s = 40 degrees

Find: a and b

sin 40o

10 m/s

10 m/s * sin = a

10 m/s10 m/s40

cos 40o

10 m/s

10 m/s * cos = b

10 m/s10 m/s40

b = 10 m/s * cos 40 = 7.66 m/sa = 10 m/s * sin 40 = 6.43 m/so o

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

Inverse Trig Functions

If sin is a trig function

then sin-1 is aninverse trig function

:inverse trig functions simply “undo” trig functions

SOHCAHTOA

• SOH Sine = Opposite/Hypotenuse

• CAH Cosine = Adjacent/Hypotenuse

• TOA Tangent = Opposite/Adjacent

Calculate the vertical (a) and horizontalsides of this right triangle.

sin 20 = a

ocos 20 =

a = 25 (sin 20)a = 8.55

b = 25 (cos 20)b = 23.49

Solve for the lengthof the hypotenuse (c)and the angle, .

c = 152 + 102

c = 325

c = 18.03

tan = 15

= tan-1 (1.5)

= 56.3

• Use the SI system– AKA Metric System– 4 basic units

• length -- meter• mass -- kilogram• time -- second• temperature -- degree Kelvin

(Celsius)

Radio Flyer

50 lbs

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

F = Fx + Fy

F = magnitude of F = 50 lbs

cos 45 =

sin 45 =

F = Fx + Fy

= cos 45

= sin 45Fy

= 50 lbs (cos 45 ) = 50 lbs * 0.707 = 35.4 lbs

= 50 lbs (sin 45 ) = 50 lbs * 0.707 = 35.4 lbs

Radio Flyer

50 lbs

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

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

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

S = 10

10cos(40)

10sin(40)

Vector Decomposition

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

R = Fm+W

• 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.

Vector Addition

• Finding the horizontal and vertical components of each vector makes it easy to find the resultant.

Vector Addition

• 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.

Vector Addition

• Use the following formulas to convert the coordinates into polar notation:

• = arctan

Vector Addition

2x sss

S2 = 3m, 165o

S1 = 6m, 40o

S2 = 3m, 165o

S1 = 6m, 40o

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

S2 = 3m, 165o

S1 = 6m, 40o

• |S| = = 4.94 m

• = arctan = 69.9o

Polar Notation

Mathematical Fundamentals

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