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

11

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”

12

Two types of TRIG problems

Type A

Type B

Given Solve For c & a & b

a & b c &

13

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

15

Inverse Trig Functions

If sin is a trig function

then sin-1 is aninverse trig function

:inverse trig functions simply “undo” trig functions

16

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

26

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

27

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

36

• 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

37

• 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