MATH and GEAS Formulas.pdf

ALGEBRA 1

Man-hours (is always assumed constant)

LOGARITHM

(Wor ker s1 )(time1 ) quantity.of .work 1

x = log b N → N = b

x

Properties

=

(Wor ker s 2 )(time 2 ) quantity.of .work 2

ALGEBRA 2

= log x + log y  x  log   = log x − log y  y  log x n = n log x

log( xy)

UNIFORM MOTION PROBLEMS

S = Vt

log x

logb x

=

log a a

=1

Traveling with the wind or downstream:

log b

V total

REMAINDER AND FACTOR THEOREMS

Traveling against the wind or upstream:

Given:

V total = V 1 − V 2

f ( x) ( x − r )

DIGIT AND NUMBER PROBLEMS

Remainder Theorem: Remainder = f(r) Factor Theorem: Remainder = zero

100h + 10t + u

QUADRATIC EQUATIONS

where:

Ax 2

+ Bx + C = 0

Root =

− B ±

= V 1 + V 2

B 2

− 4 AC

→

2-digit number

h = hundred’s digit t = ten’s digit u = unit’s digit

CLOCK PROBLEMS

2 A

Sum of the roots = - B/A Products of roots = C/A MIXTURE PROBLEMS

Quantity Analysis: A + B = C Composition Analysis: Ax + By = Cz

where:

WORK PROBLEMS Rate of doing work = 1/ time Rate x time = 1 (for a complete job) Combined rate = sum of individual rates

PDF created with pdfFactory trial version www.pdffactory.com

x = distance traveled by the minute hand in minutes x/12 = distance traveled by the hour hand in minutes

PROGRESSION PROBLEMS HARMONIC PROGRESSION (HP)

a1 an

= first term = nth term

am d

= any term before a n = common difference

S

= sum of all “n” terms

-

Mean – Mean – middle term or terms between two terms in the progression.

ARITHMETIC PROGRESSION (AP) -

difference of any 2 no.’s is constant calcu function: LINEAR (LIN)

an

= am + (n − m)d

d = a2

S =

n

S =

n

2

2

n h term Common difference

− a1 = a3 − a2 ,...etc (a1 + an )

COIN PROBLEMS Penny = Penny = 1 centavo coin Nickel = Nickel = 5 centavo coin Dime = Dime = 10 centavo coin Quarter = = 25 centavo coin Half-Dollar = = 50 centavo coin

DIOPHANTINE EQUATIONS If the number of equations is less than the number of unknowns, then the equations are called “Diophantine Equations”.

Sum of ALL terms

[2a1 + (n − 1) d ]

a sequence of number in which their reciprocals form an AP calcu function: LINEAR (LIN)

Sum of ALL terms

GEOMETRIC PROGRESSION (GP) -

RATIO of any 2 adj, terms is always constant Calcu function: EXPONENTIAL (EXP)

an

= a m r n−m

n h term

ALGEBRA 3 Fundamental Principle: “If one event can occur in m different ways, and after it has occurred in any one of these ways, a second event can occur in n different ways, and then the number of ways the two events can occur in succession is mn different mn different ways”

PERMUTATION Permutation of n objects taken r at at a time

nP r r =

S =

S =

a2 a1

=

a3

− 1)

r − 1 a1 (1 − r n ) 1 − r

(n − r )!

ratio

a2

a1 ( r n

n!

=

Permutation of n objects taken n at a time

→ r > 1 → r < 1

Sum of ALL terms, r >1

nP n

= n!

Permutation of n objects with q,r,s, etc. objects are alike

= P =

Sum of ALL terms, r < 1

n! q!r ! s!...

Permutation of n objects arrange in a circle

S =

a1 1 − r

→ r < 1 & n = ∞

Sum of ALL terms, r<1,n=


P = ( n − 1)!

PROGRESSION PROBLEMS HARMONIC PROGRESSION (HP)

a1 an

= first term = nth term

am d

= any term before a n = common difference

S

= sum of all “n” terms

-

Mean – Mean – middle term or terms between two terms in the progression.

ARITHMETIC PROGRESSION (AP) -

difference of any 2 no.’s is constant calcu function: LINEAR (LIN)

an

= am + (n − m)d

d = a2

S =

n

S =

n

2

2

n h term Common difference

− a1 = a3 − a2 ,...etc (a1 + an )

COIN PROBLEMS Penny = Penny = 1 centavo coin Nickel = Nickel = 5 centavo coin Dime = Dime = 10 centavo coin Quarter = = 25 centavo coin Half-Dollar = = 50 centavo coin

DIOPHANTINE EQUATIONS If the number of equations is less than the number of unknowns, then the equations are called “Diophantine Equations”.

Sum of ALL terms

[2a1 + (n − 1) d ]

a sequence of number in which their reciprocals form an AP calcu function: LINEAR (LIN)

Sum of ALL terms

GEOMETRIC PROGRESSION (GP) -

RATIO of any 2 adj, terms is always constant Calcu function: EXPONENTIAL (EXP)

an

= a m r n−m

n h term

ALGEBRA 3 Fundamental Principle: “If one event can occur in m different ways, and after it has occurred in any one of these ways, a second event can occur in n different ways, and then the number of ways the two events can occur in succession is mn different mn different ways”

PERMUTATION Permutation of n objects taken r at at a time

nP r r =

S =

S =

a2 a1

=

a3

− 1)

r − 1 a1 (1 − r n ) 1 − r

(n − r )!

ratio

a2

a1 ( r n

n!

=

Permutation of n objects taken n at a time

→ r > 1 → r < 1

Sum of ALL terms, r >1

nP n

= n!

Permutation of n objects with q,r,s, etc. objects are alike

= P =

Sum of ALL terms, r < 1

n! q!r ! s!...

Permutation of n objects arrange in a circle

S =

a1 1 − r

→ r < 1 & n = ∞

Sum of ALL terms, r<1,n=


P = ( n − 1)!

COMBINATION

PROBABILITY

Combination of of n objects taken r at a time

Probability of an event to occur (P)

nC r

n!

=

(n − r )!r !

P =

Combination of n objects taken n at a time

nC n

Q=1–P

Combination of of n objects taken 1, 2, 3…n at a time

−1

MULTIPLE EVENTS Mutually exclusive events without a common outcome

P A or B = P A + PB

BINOMIAL EXPANSION

Properties of a binomial expansion: (x + y)n 1.

total _ outcomes

Probability of an event not to occur (Q)

=1

C = 2 n

number _ of _ successful _ outcomes

Mutually exclusive events with a common outcome

The number of terms in the resulting resulting expansion expansion is

P A or B = P A + PB – P A&B

equal to “n+1 “n+1”” 2. The powers of x decreases by 1 in the successive terms while the powers of y increases

Dependent/Independent Probability

by 1 in the successive terms. 3.

P AandB =P A × PB

The sum of the powers powers in each term is always always equal to “n “n” n

4.

n

The first term is x while the last term in y both of the terms having a coefficient of 1. 1.

REPEATED TRIAL PROBABILITY

r

th

r term in the expansion (x + y)

P = nCr p q

n

r th term = nCr-1 (x)n-r+1 (y)r-1 term involving y r in the expansion (x + y) n r

n-r

y term = nCr (x)

p = probability that the event happen q = probability that the event failed

VENN DIAGRAMS

r

(y)

sum of coefficients of (x + y) n

Sum = (coeff. of x + coeff. of y)

n-r

n

Venn diagram in mathematics is a diagram representing a set or sets and the logical, relationships between them. The sets are drawn as circles. The method is named after the British mathematician and logician John Venn.

sum of coefficients of (x + k) n

Sum = (coeff. of x + k) n – (k)n


PLANE TRIGONOMETRY ANGLE, MEASUREMENTS & CONVERSIONS 1 revolution = 360 degrees 1 revolution = 2 ! radians 1 revolution = 400 grads

TRIGONOMETRIC IDENTITIES sin A + cos A = 1 2

1 + cot 2 A = csc2 A 1 + tan 2 A = sec 2 A sin( A ± B )

= sin A cos B ± cos A sin B cos( A ± B ) = cos A cos B msin A sin B tan A ± tan B tan( A ± B ) = cot( A ± B ) =

1 revolution = 6400 mils

cos 2 A = cos A − sin A 2

Relations between two angles (A & B) Complementary angles  A + B = 90°

tan 2 A = cot 2 A =

Supplementary angles  A + B = 180° Explementary angles  A + B = 360° Measurement

NULL

REFLEX

! = 0° 0° < ! < 90° ! = 90° 90° < ! < 180° ! =180° 180° < ! < 360°

FULL OR PERIGON

! = 360°

OBTUSE STRAIGHT

2

2 tan A 1 − tan 2 A cot 2 A − 1 2 cot A

SOLUTIONS TO OBLIQUE TRIANGLES SINE LAW

Angle (")

RIGHT

1 mtan A tan B cot A cot B m1

cot A ± cot B 2 sin A cos B

sin 2 A =

1 revolution = 6400 gons

ACUTE

2

a sin

=

b sin B

=

c sin C

COSINE LAW

a2 = b2 + c2 – 2 b c cos A b2 = a2 + c2 – 2 a c cos B c2 = a2 + b2 – 2 a b cos C

Pentagram – golden triangle (isosceles)

36°

AREAS OF TRIANGLES AND QUADRILATERALS

72° 72°

TRIANGLES 1. Given the base and height

Area

=

1 2

bh

2. Given two sides and included angle

Area =


1 2

ab sin θ

3. Given three sides 4. Quadrilateral circumscribing in a circle

Area = s ( s − a )( s − b)( s − c)

s =

Area = rs

a+b+c 2

s

4. Triangle inscribed in a circle

Area

=

abc 4r

Area = rs

6. Triangle escribed in a circle

Area

= r ( s − a )

QUADRILATERALS 1. Given diagonals and included angle

Area =

1 2

d 1d 2 sin θ

2. Given four sides and sum of opposite angles

θ

= ( s − a)( s − b)( s − c )( s − d ) − abcd cos 2 θ A + C B + D = =

s

=

Area

2 2 a + b + c + d 2

3. Cyclic quadrilateral – is a quadrilateral inscribed in a circle

s

=

r =

=

THEOREMS IN CIRCLES

5. Triangle circumscribing a circle

Area =

= abcd a + b + c + d

Area

( s − a)( s − b)( s − c )( s − d )

a + b + c + d 2 ( ab + cd )( ac + bd )(ad + bc) 4( Area)

d 1 d 2= ac + bd → Ptolemy’s Theorem


2

SIMILAR TRIANGLES A1 A2

2

2

Number of diagonal lines (N): 2

A   B   C   H  =    =   =   =    a   b   c   h 

SOLID GEOMETRY POLYGONS

3 sides – Triangle 4 sides – Quadrilateral/Tetragon/Quadrangle 5 sides – Pentagon 6 sides – Hexagon 7 sides – Heptagon/Septagon 8 sides – Octagon 9 sides – Nonagon/Enneagon 10 sides – Decagon 11 sides – Undecagon 12 sides – Dodecagon 15 sides – Quidecagon/ Pentadecagon 16 sides – Hexadecagon 20 sides – Icosagon 1000 sides – Chillagon

n

N =

2

2

( n − 3)

Area of a regular polygon inscribed in a circle of radius r 1  360°  Area = nr 2 sin   2  n  Area of a regular polygon circumscribing a circle of radius r  180°  2 Area = nr tan    n  Area of a regular polygon having each side measuring x unit length

Area

1

=

4

 180°    n 

nx 2 cot 

PLANE GEOMETRIC FIGURES CIRCLES

A =

Let: n = number of sides  = interior angle ! = exterior angle

πd 2

= πr 2

4 Circumference = πd = 2πr

Sector of a Circle

Sum of interior angles:

A =

S = n  = (n – 2) 180°

1 2

rs

=

1 2

r 2θ

2

A =

Value of each interior angle

θ

=

(n − 2)(180°) n

= 180° − θ =

360°

= r θ ( rad ) =

πr θ (deg) 180°

Segment of a Circle

Value of each exterior angle

α

s

πr θ (deg)

A

360° n

A sector – A triangle

ELLIPSE

Sum of exterior angles:

S = n " = 360°

segment =

A =  a b PARABOLIC SEGMENT

A =


2 3

bh

PYRAMID TRAPEZOID

1

A =

2

( a + b) h

PARALLELOGRAM

A = ab sin α

1 V = Bh 3 A( lateral ) = ∑ A faces A( surface) = A( lateral ) + B

A = bh A =

1 2

Frustum of a Pyramid

d 1d 2 sin θ

V =

RHOMBUS

A =

1 2

d 1d 2

3

( A1 + A2 + A1 A2 )

A1 = area of the lower base A2 = area of the upper base

= ah

= a 2 sin α

h

PRISMATOID

V = SOLIDS WITH PLANE SURFACE

h 6

( A1 + A2

Lateral Area = (No. of Faces) (Area of 1 Face)

Am = area of the middle section

Polyhedron – a solid bounded by planes. The bounding planes are referred to as the faces and the intersections of the faces are called the edges. The intersections of the edges are called vertices.

REGULAR POLYHEDRON

PRISM

a solid bounded by planes whose faces are congruent regular polygons. There are five regular polyhedrons namely: A. B. C. D. E.

V = Bh

+ 4 Am )

A(lateral) = PL A(surface) = A(lateral) + 2B where: P = perimeter of the base L = slant height B = base area

Truncated Prism

 ∑ heights     number of heights 

V = B


Tetrahedron Hexahedron (Cube) Octahedron Dodecahedron Icosahedron

e m a N

n o r d e h a r t e T

n o r d e h a x e H

f e o E l g e C n p A i a y r T F T

e r a u q S

f S o E . o C 4 N A F

6

f S o E . 6 o G N D E

2 1

S f E o C . I o T N R E V

n o r d e h a t c O

n o r d e h a c e d o D

n o r d e h a s o c I

e l g n a i r T

n o g a t n e P

e l g n a i r T

8

2 1

0 2

2 1

0 3

FRUSTUM OF A CONE

V =

h 3

( A1 + A2

+

A1 A2

= π ( R + r ) L

A( lateral )

SPHERES AND ITS FAMILIES SPHERE

V =

4 3

πr 3

A( surface )

0 3

= 4πr 2

SPHERICAL LUNE 4

8

6

0 2

2 1

is that portion of a spherical surface bounded by the halves of two great circles 2

r o f E s M x a l U 2 2 1 u L = m O r o V V F 3

3 3

x

x

2

=

V

3

3

x

6 6 . 7

3

x

=

=

=

V

V

Where: x = length of one edge

SOLIDS WITH CURVED SURFACES

V = Bh = KL A(lateral) = PkL = 2  r h A(surface) = A(lateral) + 2B Pk = perimeter of right section K = area of the right section B = base area L= slant height

90°

8 1 . 2

V

CYLINDER

=

A( surface )

πr θ (deg)

SPHERICAL ZONE is that portion of a spherical surface between two parallel planes. A spherical zone of one base has one bounding plane tangent to the sphere.

= 2π

A( zone )

r h

SPHERICAL SEGMENT is that portion of a sphere bounded by a zone and the planes of the zone’s bases. 2

V =

πh

V =

πh

V =

3 6 πh 6

(3r − h)

(3a 2

+ h2 )

(3a 2

+ 3b 2 + h 2 )

CONE

1 V = Bh 3 A(lateral ) = πrL

SPHERICAL WEDGE is that portion of a sphere bounded by a lune and the planes of the half circles of the lune. 3

V =


πr θ (deg) 270°

SPHERICAL CONE

PARABOLOID

is a solid formed by the revolution of a circular sector about its one side (radius of the circle).

a solid formed by rotating a parabolic segment about its axis of symmetry.

V =

1 3

V =

A( zone ) r

A( surface)

= A( zone ) + A(lateralofcone )

SPHERICAL PYRAMID is that portion of a sphere bounded by a spherical polygon and the planes of it s sides. 3

πr E

V =

540°

E = [(n-2)180°]

1 2

πr 2 h

SIMILAR SOLIDS 3

3

3

H   R   L  =    =   =   V 2  h   r   l  2 2 2 A1  H   R   L  =   =   =   A2  h   r   l  2 3  V 1   A1     =  A   V  2   2  V 1

E = Sum of the angles E = Spherical excess n = Number of sides of the given spherical polygon

ANALYTIC GEOMETRY 1

SOLIDS BY REVOLUTIONS RECTANGULAR COORDINATE SYSTEM TORUS (DOUGHNUT) a solid formed by rotating a circle about an axis not passing the circle.

V = 22Rr 2 A(surface) = 4 2Rr

4 3

Distance between two points

d =

ELLIPSOID

V =

x = abscissa y = ordinate

πabc

( x2 − x1 ) 2

Slope of a line

m

OBLATE SPHEROID

= tan θ =

a solid formed by rotating an ellipse about its minor axis. It is a special ellipsoid with

V =

4 3

c=a

− y1 x2 − x1

y2

Division of a line segment

+ x 2 r 1 x = r 1 + r 2 x1 r 2

2

πa b

+ ( y2 − y1 )2

y

=

+ y 2 r 1 r 1 + r 2

y1 r 2

PROLATE SPHEROID a solid formed by rotating an ellipse about its major axis. It is a special ellipsoid with

V =

4 3

πab

2

c=b

Location of a midpoint

x =


x1 + x 2 2

y =

y1 + y 2 2

STRAIGHT LINES

Distance between two parallel lines

d

General Equation

=

Ax + By + C = 0

A2 + B 2

Slope relations between parallel lines: m1 = m2

Point-slope form

y – y1 = m(x – x1) Two-point form

y − y1

C 1 − C 2

=

− y1 ( x − x1 ) x2 − x1

y2

Slope and y-intercept form

Line 1  Ax + By + C 1 = 0 Line 2  Ax + By + C 2 = 0 Slope relations between perpendicular lines: m1m2 = –1

Line 1  Ax + By + C 1 = 0 Line 2  Bx – Ay + C 2 = 0

y = mx + b PLANE AREAS BY COORDINATES Intercept form

x a

+

y b

=1

Slope of the line, Ax + By + C = 0

m=

−

A B

Angle between two lines

θ

 m − m  = tan −1  2 1    1 + m1m2 

Note: Angle ! is measured in a counterclockwise direction. m2 is the slope of the terminal side while m1 is the slope of the initial side.

Distance of point (x 1,y1) from the line Ax + By + C = 0;

d =

A =

2 y1 , y2 , y3 ,.... yn , y1

Note: The points must be arranged in a counter clockwise order.

LOCUS OF A MOVING POINT The curve traced by a moving point as i t moves in a plane is called the locus of the point.

SPACE COORDINATE SYSTEM Length of radius vector r:

r = x 2

+ y 2 + z 2

Distance between two points P 1(x1,y1,z1) and P2(x2,y2,z2)

d =

Ax1 + By1 + C

±

1 x1 , x2 , x3 ,.... xn , x1

A2 + B 2

Note: The denominator is given the sign of B


( x2

− x1 )2 + ( y2 − y1 )2 + ( z 2 − z 1 ) 2

Standard Equation:

ANALYTIC GEOMETRY 2

(x – h)2 + (y – k)2 = r 2 General Equation:

CONIC SECTIONS a two-dimensional curve produced by slicing a plane through a three-dimensional right circular conical surface

x2 + y2 + Dx + Ey + F = 0 Center at (h,k):

Ways of determining a Conic Section

h=−

1. By Cutting Plane 2. Eccentricity 3. By Discrimination 4. By Equation

D 2

;

k = −

or

r =

r 2

= h 2 + k 2 −

F

Ax2 + Cy2 + Dx + Ey + F = 0 **

Circle Parabola

Parallel to base

Eccentricity e

# 0

Parallel to element

e = 1.0

none

e < 1.0

Parallel to axis

e > 1.0

Discriminant

Equation**

Circle

B - 4AC < 0, A = C

A=C

Parabola

B - 4AC = 0

Ellipse

B - 4AC < 0, A $ C

Ellipse Hyperbola

2

2

2

2

Hyperbola B - 4AC > 0

2

Radius of the circle:

General Equation of a Conic Section:

Cutting plane

E

A $ C same sign

2

D 2 + E 2

− 4 F

PARABOLA a locus of a moving point which moves so that it’s always equidistant from a fixed point called focus and a fixed line called directrix.

where: a = distance from focus to vertex = distance from directrix to vertex AXIS HORIZONTAL:

Sign of A opp. of B A or C = 0

1

Cy2 + Dx + Ey + F = 0 Coordinates of vertex (h,k):

k = −

CIRCLE A locus of a moving point which moves so that its distance from a fixed point called the center is constant.

E 2C

substitute k to solve for h Length of Latus Rectum:

LR =


D C

AXIS VERTICAL:

ELLIPSE

2

Ax + Dx + Ey + F = 0 Coordinates of vertex (h,k):

=−

h

D

a locus of a moving point which moves so that the sum of its distances from two fixed points called the foci is constant and is equal to the length of its major axis. d = distance of the center to the directrix

2 A

substitute h to solve for k

Length of Latus Rectum:

LR =

E

STANDARD EQUATIONS:

A

Major axis is horizontal:

( x − h) 2

STANDARD EQUATIONS:

+

a2

( y − k ) 2 b2

=1

Opening to the right: Major axis is vertical: 2

( x − h) 2

(y – k) = 4a(x – h)

b

Opening to the left:

(y – k)2 = –4a(x – h)

2

+

( y − k ) 2 a

2

General Equation of an Ellipse:

Ax2 + Cy2 + Dx + Ey + F = 0

Opening upward:

(x – h) 2 = 4a(y – k) Opening downward:

Coordinates of the center:

h

=−

2

(x – h) = –4a(y – k) Latus Rectum (LR) a chord drawn to the axi s of symmetry of the curve.

LR= 4a for a parabola Eccentricity (e) the ratio of the distance of the moving point f rom the focus (fixed point) to i ts distance from the directrix (fixed line).

D 2

; k =

−

E 2C

If A > C, then: a2 = A; b 2 = C If A < C, then: a2 = C; b2 = A KEY FORMULAS FOR ELLIPSE Length of major axis:

2a

Length of minor axis:

2b

Distance of focus to center :

e=1

=1

for a parabola

c

=


a2

− b2

Length of latus rectum: 2

2b

=

LR

(y – k) = m(x – h)

a

Transverse axis is horizontal:

m=

Eccentricity:

c

e=

a

a

=

Equation of Asymptote:

±

b a

Transverse axis is vertical:

d

m=

HYPERBOLA a locus of a moving point which moves so that the difference of its distances from two fixed points called the foci is constant and is equal to l ength of its transverse axis.

±

a b

KEY FORMULAS FOR HYPERBOLA

2a

Length of transverse axis: Length of conjugate axis:

2b

Distance of focus to center :

c= d = distance from center to directrix a = distance from center to vertex c = distance from center to focus

Length of latus rectum:

LR

STANDARD EQUATIONS Transverse axis is horizontal

( x − h) a

2

2

−

( y − k ) b

2

2

a 2 + b2

=

2b 2 a

Eccentricity:

=1

e

c

a

a

d

= =

Transverse axis is vertical:

( y − k ) 2 a

2

−

( x − h ) 2 b

2

=1

POLAR COORDINATES SYSTEM x = r cos 

GENERAL EQUATION 2

2

Ax – Cy + Dx + Ey + F = 0 Coordinates of the center:

h=−

D 2

;

k = −

E 2C

If C is negative, then: a2 = C, b 2 = A If A is negative, then: a2 = A, b 2 = C


y = r sin 

r = x 2 tan θ

+ y 2

=

y x

SPHERICAL TRIGONOMETRY Important propositions 1. If two angles of a spherical triangle are equal, the sides opposite are equal; and conversely.

Napier’s Rules 1. The sine of any middle part is equal to the product of the cosines of the opposite parts.

Co-op 2. The sine of any middle part is equal to the product of the tangent of the adjacent parts.

Tan-ad

2. If two angels of a spherical triangle are unequal, the sides opposite are unequal, and the greater side lies opposite the greater angle; and conversely.

Important Rules:

3. The sum of two sides of a spherical triangle is greater than the third side.

2. When the hypotenuse of a right spherical triangle is less than 90°, the two legs are of the same quadrant and conversely.

a+b>c 4. The sum of the sides of a spherical triangle is less than 360°.

1. In a right spherical triangle and oblique angle and the side opposite are of the same quadrant.

3. When the hypotenuse of a right spherical triangle is greater than 90°, one leg is of the first quadrant and the other of the second and conversely.

0° < a + b + c < 360° QUADRANTAL TRIANGLE 5. The sum of the angles of a spherical triangle is greater that 180° and less than 540° .

180° < A + B + C < 540° 6. The sum of any two angles of a spherical triangle is less than 180° plus the third angle.

is a spherical triangle having a side equal to 90°.

SOLUTION TO OBLIQUE TRIANGLES Law of Sines:

sin a

A + B < 180° + C SOLUTION TO RIGHT TRIANGLES NAPIER CIRCLE Sometimes called Neper’s circle or Neper’s pentagon, is a mnemonic aid to easily find all relations between the angles and sides in a right spherical triangle.

sin A

=

sin b sin B

=

sin c sin C

Law of Cosines for sides:

= cos b cos c + sin b sin c cos A cos b = cos a cos c + sin a sin c cos B cos c = cos a cos b + sin a sin b cos C cos a

Law of Cosines for angles:

cos A =

− cos B cos C + sin B sin C cos a cos B = − cos A cos C + sin A sin C cos b cos C = − cos A cos B + sin A sin B cos c


AREA OF SPHERICAL TRIANGLE

A =

π R 2 E 180°

R = radius of the sphere E = spherical excess in degrees,

E = A + B + C – 180°

Derivatives dC dx d dx d dx

=0

(u + v ) = (uv) = u v

d  u 

TERRESTRIAL SPHERE

d dx

Prime meridian (Longitude = 0°)

d

(u n )

u

=

d  c 

Latitude = 0° to 90°

  = dx  u 

Longitude = 0° to +180° (eastward) = 0° to –180° (westward)

d dx d dx d dx

1 statute mile = 8 furlongs = 80 chains

dx

+v

dx du

−u v

= nu n −1

Equator (Latitude = 0°)

1 statute mile = 5280 feet = 1760 yards

dx du

dv

+

dx dv dx

2

du dx

du dx

1 nautical mile = 6080 feet = 1852 meters

dx dv

  = dx  v 

Radius of the Earth = 3959 statute miles

1 min. on great circle arc = 1 nautical mile

du

dx 2 u

−c u

du dx 2

( a ) = a ln a u

u

(e u ) = e u

(ln a u ) =

du dx

du dx log a e

du dx

u

du d

(ln u ) = dx dx u d du (sin u ) = cos u dx dx d du (cos u ) = − sin u dx dx d du 2 (tan u ) = sec u dx dx d du (cot u ) = − csc 2 u dx dx d du (sec u ) = sec u tan u dx dx d du (csc u ) = − csc u cot u dx dx d 1 du (sin −1 u ) = 2 dx 1 − u dx


d dx d dx d dx d dx d dx d dx d dx d

−1

(cos u ) = (tan −1 u ) = −1

(cot u ) = (sec −1 u ) = (csc −1 u ) =

−1 1− u2 1

du

DIFFERENTIAL CALCULUS

dx

du

1 + u dx − 1 du 2

1 + u dx 1 du 2

− 1 dx

u u2

−1 u u

2

(sinh u ) = cosh u (cosh u ) = sinh u

du

− 1 dx

LIMITS Indeterminate Forms

0 0

,

dx du 2 (tanh u ) = sec h u dx dx d du (coth u ) = − csc h 2 u dx dx d du (sec hu ) = − sec hu tanh u dx dx d du (csc hu ) = − csc hu coth u dx dx d 1 du (sinh −1 u ) = 2 dx u + 1 dx d 1 du (cosh −1 u ) = 2 dx u − 1 dx d 1 du (tanh −1 u ) = dx 1 − u 2 dx − 1 du d (sinh −1 u ) = 2 dx u − 1 dx d − 1 du −1 (sec h u ) = dx u 1 − u 2 dx − 1 du d −1 (csc h u ) = dx u 1 + u 2 dx

(0)(∞),

∞ - ∞,

00 ,

∞0 ,

1∞

L’Hospital’s Rule

du dx du

∞ , ∞

Lim x → a

f ( x) g ( x )

= Lim x → a

f ' ( x) g ' ( x )

= Lim x → a

f " ( x) g " ( x )

.....

Shortcuts Input equation in the calculator TIP 1: if x # 1, substitute x = 0.999999 TIP 2: if x #

∞ , substitute x = 999999

TIP 3: if Trigonometric, convert to RADIANS then do tips 1 & 2

MAXIMA AND MINIMA Slope (pt.) Y’ MAX 0

Y” (-) dec

Concavity down

MIN

0

(+) inc

up

INFLECTION

-

No change

-

HIGHER DERIVATIVES th n n derivative of x

d n dx n nth derivative of xe

d n dx n PDF created with pdfFactory trial version www.pdffactory.com

( x n ) = n ! n

( xe n ) = ( x + n) e X

TIME RATE the rate of change of the v ariable with respect to time

+

dx

−

dx

dt

dt

= increasing

rate

−1

= decreasing rate

APPROXIMATION AND ERRORS If “dx” is the error in the measurement of a quantity x, then “dx/x” is called the RELATIVE ERROR.

3

[1 + ( y ' ) ] 2 2

=

y"

INTEGRAL CALCULUS 1 ∫ du = u + C ∫ adu = au + C ∫ [ f (u) + g (u)]du = ∫ f (u)du + ∫ g (u)du u u du = + C ..............(n ≠ 1) ∫ n +1 du ∫ u = ln u + C n +1

n

a

u

u

2

∫ a ∫ u

2

du

+ u2

2

=

du u

2

−a

2

1 a

tan

=

1 a

−1

u a

+ C

sec −1

u a

+ C

u  −1  cos 1 = −   + C ∫ 2au − u 2  a  ∫ sinh udu = cosh u + C

∫ cosh udu = sinh u + C ∫ sec h udu = tanh u + C ∫ csc h udu = − coth u + C ∫ sec hu tanh udu = − sec hu + C ∫ csc hu coth udu = − csc hu + C ∫ tanh udu = ln cosh u + C ∫ coth udu = ln sinh u + C du u sinh = + C ∫ u + a a 2

2

−1

2

∫ u

u

∫ a du = ln a + C ∫ e du = e + C ∫ sin udu = − cos u + C ∫ cos udu = sin u + C ∫ sec udu = tan u + C ∫ csc udu = − cot u + C ∫ sec u tan udu = sec u + C ∫ csc u cot udu = − csc u + C u

2

du

RADIUS OF CURVATURE

R

∫ tan udu = ln sec u + C ∫ cot udu = ln sin u + C ∫ sec udu = ln sec u + tan u + C ∫ csc udu = ln csc u − cot u + C du u sin = + C ∫ a − u a

2

du 2

u

− a2 du

∫ u

u2 du

+ a2

= cosh −1 + C a

=− 1

1 a

∫ a − u a du 1 = ∫ a − u a coth ∫ udv = uv −∫ vdu 2

2

2

2

2


=

sinh

tanh −1 −1

u a u a

−1

a u

+ C

+ C .............. u < a + C .............. u > a

PLANE AREAS

CENTROIDS

Plane Areas bounded by a curve and the coordinate axes:

Half a Parabola

x 2

∫

A = y ( curve) dx

x

3

=

8 2

x1 y 2

y =

∫

A = x( curve) dy

x 2

∫

A = ( y( up )

− y( down ) )dx

x1

Whole Parabola

y

∫

A = ( x( right ) − x(left ) )dy y1

Plane Areas bounded by polar curves:

A =

2

=

h

5

Triangle

y 2

1

h

5

y1

Plane Areas bounded by a curve and the coordinate axes:

b

θ2

∫

2 θ1

x

=

y

=

1 3 1

b=

3

r 2 d θ

h=

2 3 2 3

b h

LENGTH OF ARC

CENTROID OF PLANE AREAS (VARIGNON’S THEOREM)

Using a Vertical Strip: x 2

∫

A • x = dA • x x1

x 2

A • y

y

x1

2

Using a Horizontal Strip:

A • x =

x

∫ dA • 2

2

y 2

 dx  S = ∫ 1 +    dy dy   y 1

z 2

S =

∫

z 1

y1

y 2

A • y

2

 dy  S = ∫ 1 +   dx  dx  x 1

= ∫ dA • y 2

x 2

= ∫ dA • y y1


2

 dx     dz 

2

dy  +    dz  dz 

INTEGRAL CALCULUS 2

CENTROIDS OF VOLUMES x 2

∫

V • x = dV • x x1

y 2

TIP 1: Problems will usually be of this nature: • “Find the area bounded by” • “Find the area revolved around..”

V • y = dV • y

TIP 2: Integrate only when the shape is IRREGULAR, otherwise use the prescribed f ormulas

WORK BY INTEGRATION Work = force × distance

∫

y1

x 2

VOLUME OF SOLIDS BY REVOLUTION

y 2

∫

∫

W = Fdx = Fdy ; where F = k x

Circular Disk Method

x1

y1

x2

∫

V = π R 2 dx

Work done on spring

W =

x1

Cylindrical Shell Method y2

∫

V = 2π RL dy

1 2

k ( x2

2

− x12 )

k = spring constant x1 = initial value of elongation x2 = final value of elongation

Work done in pumping liquid out of the container at its top

y1

Circular Ring Method x2

∫

V = π ( R

2

− r )dx

Work = (density)(volume)(distance)

2

x1

PROPOSITIONS OF PAPPUS First Proposition: If a plane arc is revolved about a coplanar axis not crossing the arc, the area of the surface generated is equal to the product of the length of the arc and the circumference of the circle described by the centroid of the arc.

A = S • 2π r

∫

A = dS • 2π r Second Proposition: If a plane area is revolved about a coplanar axis not crossing the area, the volume generated is equal to the product of the area and the circumference of the circle described by the centroid of the area.

V = A • 2π r

∫

V = dA • 2π r

Force = (density)(volume) = #v Specific Weight:

γ

=

Weight Volume

!water = 9.81 kN/m 2 SI !water = 45 lbf/ft 2 cgs Density:

ρ

=

mass Volume

#water = 1000 kg/m 3 SI #water = 62.4 lb/ft 3 cgs #subs = (substance) (#water ) 1 ton = 2000lb


MOMENT OF INERTIA Moment of Inertia about the x- axis: x 2

= ∫ y

I x

2

dA

x1

Ellipse

I x

=

I y

=

Moment of Inertia about the y- axis:

πab 3 4 πa 3b 4

y 2

= ∫ x 2 dA

I y

FLUID PRESSURE

F = wh A = γ h A

y1

Parallel Axis Theorem The moment of inertia of an area with respect to any coplanar line equals the moment of inertia of the area with respect to the parallel centroidal line plus the area times the square of the distance between the lines.

I x

= Ixo = Ad 2

∫

F = wh dA F = force exerted by the fluid on one side of the area h = distance of the c.g. to the surface of liquid w = specific weight of the liquid ( $) A = vertical plane area

Moment of Inertia for Common Geometric Figures

Specific Weight:

Square

γ

I x

bh3

=

3 bh

=

I xo

=

Triangle

I x

=

3

12 bh3 36

I xo =

Circle

I xo

=

4

πr 4

Half-Circle

I x

=

πr 4 8

Quarter-Circle

I x

=

Volume

!water = 9.81 kN/m 2 SI !water = 45 lbf/ft 2 cgs

3

12

bh

Weight

πr 4 16


MECHANICS 1

CABLES PARABOLIC CABLES the load of the cable of distributed horizontally along the span of the cable.

VECTORS Dot or Scalar product

Uneven elevation of supports

P • Q = P Q cosθ

P • Q

= P x Q x + P y Q y + P z Q z

H =

wx1

2

2d 1

wx2

=

2

2d 2

Cross or Vector product P × Q P × Q

= P Q sin θ i

j

k

= P x

P y

P z

Q x

Q y

Q z

EQUILIBRIUM OF COPLANAR FORCE SYSTEM Conditions to attain Equilibrium:

∑ F ∑ F ∑ M

=0 ( y − axis ) = 0 ( po int) = 0 ( x − axis )

Friction

Ff = %N tan" = # " = angle of friction if no forces are applied except for the weight,

"

=

$

T 1

=

( wx1 )

T 2

=

(wx2 ) 2

2

+ H 2 + H 2

Even elevation of supports

L d

> 10

H =

wL2 8d 2

 wL  + H 2 T =    2  S = L +

8d 2 3 L

−

32d 4 5 L3

L = span of cable d = sag of cable T = tension of cable at support H = tension at lowest point of cable w = load per unit length of span S = total length of cable


CATENARY

MECHANICS 2

the load of the cable is distributed along the entire length of the cable.

Uneven elevation of supports

= wy1 T 2 = wy 2 H = wc 2 2 2 y1 = S 1 + c 2 2 2 y 2 = S 2 + c  S + y1  x1 = c ln 1   c   S + y 2  x 2 = c ln 2   c  Span = x1 + x2 T 1

RECTILINEAR MOTION Constant Velocity

S = Vt Constant Acceleration: Horizontal Motion

S = V 0 t ±

2 V = V 0 ± at V 2

Total length of cable = S 1 + S2 Even elevation of supports

T = wy

= S + c  S + y  x = c ln   c   Span = 2 x y

at 2

= V 0 2 ± 2aS

+ (sign) = body is speeding up – (sign) = body is slowing down Constant Acceleration: Vertical Motion

H = wc 2

1

2

2

Total length of cable = 2S

± H = V 0t −

1 2

gt 2

V = V 0 ± gt V 2

= V 0 2 ± 2 gH

+ (sign) = body is moving down – (sign) = body is moving up Values of g,

general estimate exact


SI (m/s 2) 9.81 9.8 9.806

English (ft/s2) 32.2 32 32.16

Variable Acceleration

V = a=

dS

ROTATION (PLANE MOTION)

dt dV

Relationships between linear & angular parameters:

dt

V = r ω

a = r α

PROJECTILE MOTION

V = linear velocity % = angular velocity (rad/s) a = linear acceleration & = angular acceleration (rad/s 2) r = radius of the flywheel Linear Symbol

Angular Symbol

S V A t

! ' " t

Distance Velocity Acceleration Time Constant Velocity

x = (V 0 cos θ )t

± y = (V 0 sin θ )t − ± y = x tan θ −

1 2

gx 2

 = "t

gt 2

Constant Acceleration

2 2

2V 0 cos θ

Maximum Height and Horizontal Range 2

y =

2

V 0 sin θ 2 g

αt 2

2 ω = ω 0 ± αt ω2

= ω 0 2 ± 2αθ

D’ALEMBERT’S PRINCIPLE

g

Maximum Horizontal Range

Assume: Vo = fixed ! = variable

=

1

+ (sign) = body is speeding up – (sign) = body is slowing down

V 0 sin 2θ

Rmax

= ω 0 t ±

max ht

2

x =

θ

V 0

2

g

⇔ θ = 45°

“Static conditions maybe produced in a body possessing acceleration by the addition of an imaginary force called reverse effective force (REF) whose magnitude is (W/g)(a) acting through the center of gravity of the body, and parallel but opposite in direction to the acceleration.”

REF = ma


 W  =   a   g 

UNIFORM CIRCULAR MOTION motion of any body moving in a circle with a constant speed.

F c ac

= =

mV 2 r V

=

WV 2

BOUYANCY A body submerged in fluid is subjected by an unbalanced force called buoyant force equal to the weight of the displaced fluid

Fb = W

gr

2

Fb = !Vd

r

Fc = centrifugal force V = velocity m = mass W = weight r = radius of track ac = centripetal acceleration g = standard gravitational acceleration

BANKING ON HI-WAY CURVES

Fb = buoyant force W = weight of body or fluid # = specific weight of fluid Vd = volume displaced of fluid or volume of submerged body

Specific Weight:

γ

=

Weight Volume

!water = 9.81 kN/m 2 SI !water = 45 lbf/ft 2 cgs Ideal Banking: The road is frictionless

tan θ

=

V 2

Non-ideal Banking: With Friction on the road V 2 tan φ = µ tan(θ + φ ) = ; gr V = velocity r = radius of track g = standard gravitational acceleration  = angle of banking of the road φ = angle of friction ' = coefficient of friction

Conical Pendulum

T = W sec !

tan θ

f =

=

F W

=

ENGINEERING MECHANICS 3

gr

IMPULSE AND MOMENTUM Impulse = Change in Momentum

F ∆t = mV − mV 0 F = force t = time of contact between the body and the force m = mass of the body V0 = initial velocity V = final velocity

Impulse, I

V 2 gr

1

g

2π

h

Momentum, P

frequency


I = F ∆t P = mV

Work, Energy and Power LAW OF CONSERVATION OF MOMENTUM “In every process where the velocity is changed, the momentum lost by one body or set of bodies is equal to the momentum gain by another body or set of bodies”

Work

W = F ⋅ S

Force Newton (N) Dyne

Distance Meter Centimeter

Work Joule ft-lb f

Pound (lbf )

Foot

erg

Potential Energy

PE = mgh Momentum lost = Momentum gained

m1V 1

Kinetic Energy

+ m2V 2 = m V + m2V ' 1 1

= Wh

' 2

=

KE linear

1 2

mV 2

1 KE rotational = I ω 2  V = r % 2

m1 = mass of the first body m2 = mass of the second body V1 = velocity of mass 1 before the impact V2 = velocity of mass 2 before the impact V1’ = velocity of mass 1 after the impact V2’ = velocity of mass 2 after the impact

I = mass moment of inertia % = angular velocity

Coefficient of Restitution (e) Mass moment of inertia of rotational INERTIA for common geometric figures:

− V 1' e= V 1 − V 2 V 2'

Type of collision

ELASTIC INELASTIC PERFECTLY INELASTIC

e

100% conserved Not 100% conserved Max Kinetic Energy Lost

Solid sphere: I = Kinetic Energy

0 < e

>1

e=0

e =1

5

2

mr

Thin-walled hollow sphere: I = Solid disk: I =

1 2

Solid Cylinder: I

Special Cases

2

2

=

1 2

m = mass of the body r = radius

hr hd

e = cot θ tan β


3

mr 2

mr

Hollow Cylinder: I =

e =

2

mr 2 1 2

2 2 m(r outer ) − r inner

Latent Heat is the heat needed by the body to change

POWER

its phase without changing its temperature.

rate of using energy

P =

W t

= F ⋅ V

Q = ±mL

1 watt = 1 Newton-m/s

Q = heat needed to change phase m = mass L = latent heat (fusion/vaporization) (+) = heat is entering (substance melts) (–) = heat is leaving (substance freezes)

1 joule/sec = 107 ergs/sec 1 hp = 550 lb-ft per second = 33000 lb-ft per min = 746 watts

Latent heat of Fusion – solid to liquid Latent heat of Vaporization – liquid to gas

LAW ON CONSERVATION OF ENERGY “Energy cannot be created nor destroyed, but it can be change from one form to another”

Values of Latent heat of Fusion and Vaporization,

Kinetic Energy = Potential Energy WORK-ENERGY RELATIONSHIP The net work done on an object always produces a change in kinetic energy of the object. Work Done =

(KE

Positive Work – Negative Work =

(KE

Total Kinetic Energy = linear + rotation HEAT ENERGY AND CHANGE IN PHASE Sensible Heat is the heat needed to change the temperature of the body without changing its phase.

Q = mc (T

Lf = 144 BTU/lb Lf = 334 kJ/kg Lf ice = 80 cal/gm Lv boil = 540 cal/gm Lf = 144 BTU/lb = 334 kJ/kg Lv = 970 BTU/lb = 2257 kJ/kg 1 calorie = 4.186 Joules 1 BTU = 252 calories = 778 ft–lbf LAW OF CONSERVATION OF HEAT ENERGY When two masses of different temperatures are combined together, the heat absorbed by the lower temperature mass is equal to the heat given up by the higher temperature mass.

Q = sensible heat m = mass c = specific heat of the substance (T = change in temperature Specific heat values

Cwater = 1 BTU/lb–°F Cwater Cwater Cice Csteam

= 1 cal/gm–°C = 4.156 kJ/kg = 50% Cwater = 48% Cwater


Heat gained = Heat lost

THERMAL EXPANSION For most substances, the physical size increase with an increase in temperature and decrease with a decrease in temperature.

(L = L&(T (L = change in length L = original length

& = coefficient of linear expansion (T = change in temperature

(V = V)(T (V = change in volume

THERMODYNAMICS In thermodynamics, there are four laws of very general validity. They can be applied to systems about which one knows nothing other than the balance of energy and matter transfer. ZEROTH LAW OF THERMODYNAMICS stating that thermodynamic equilibrium is an equivalence relation. If two thermodynamic systems are in thermal equilibrium with a third, they are also in thermal equilibrium with each other. FIRST LAW OF THERMODYNAMICS about the conservation of energy

V = original volume

) = coefficient of volume expansion

(T = change in temperature Note: In case ) is not given;

) = 3&

The increase in the energy of a closed system is equal to the amount of energy added to the system by heating, minus the amount lost in the form of work done by the system on its surroundings. SECOND LAW OF THERMODYNAMICS about entropy

The total entropy of any isolated thermodynamic system tends to increase over time, approaching a maximum value. THIRD LAW OF THERMODYNAMICS, about absolute zero temperature

As a system asymptotically approaches absolute zero of temperature all processes virtually cease and the entropy of the system asymptotically approaches a minimum value. This law is more clearly stated as: "the entropy of a perfectly crystalline body at absolute zero temperature is zero."


STRENGTH OF MATERIALS

&

SI N/m2 = Pa

SIMPLE STRESS

Stress =

Units of

Force

mks/cgs

English

Kg/cm2

lbf /m2 = psi

kN/m2 = kPa

103 psi = ksi

Axial Stress

MN/m2 = MPa

103 lbf = kips

the stress developed under the action of the f orce acting axially (or passing the centroid) of the resisting area.

GN/m2 = Gpa

σ axial

=

rea

P axial

2

N/mm = MPa

Standard Temperature and Pressure (STP) Paxial * Area

+axial = axial/tensile/compressive stress P = applied force/load at centroid of x’sectional area A = resisting area (perpendicular area)

101.325 kPa

Shearing stress the stress developed when the force is applied parallel to the resisting area.

σ s =

14.7 psi 1.032 kgf /cm2 780 torr 1.013 bar 1 atm 780 mmHg 29.92 in

P

Thin-walled Pressure Vessels

A

A. Tangential stress

Pappliedl , Area

+s = shearing stress P = applied force or load A = resisting area (sheared area)

σ T

=

the stress developed in the area of contact (projected area) between two bodies.

ρ r t

=

ρ D 2t

B. Longitudinal stress (also for Spherical)

Bearing stress

σb

= = = = = = =

σ L

=

ρ r 2t

=

ρ D 4t

= P = P A

dt

P * Abaering

+b = bearing stress P = applied force or load A = projected area (contact area) d,t = width and height of contact, respectively

+T = tangential/circumferential/hoop stress +L = longitudinal/axial stress, used in spheres r = outside radius D = outside diameter - = pressure inside the tank t = thickness of the wall F = bursting force


SIMPLE STRAIN / ELONGATION

Types of elastic deformation:

Strain – ratio of elongation to original length

a. Due to axial load HOOKE’S LAW ON AXIAL DEFORMATION “Stress is proportional to strain”

σαε

= Y ε Young ' s Modulus of Elasticity σ = E ε Modulus of Elasticity σ s = E s ε s Modulus in Shear σ V = E V ε V Bulk Modulus of Elasticity σ

1 E v

compressibility

δ

ε . = strain

=

δ L

$ = elongation L = original length Elastic Limit – the range beyond which the material WILL NOT RETURN TO ITS ORIGINAL SHAPE when unloaded but will retain a permanent deformation Yield Point – at his point there is an appreciable elongation or yielding of the material without any corresponding increase in load; ductile materials and continuous deformation Ultimate Strength – it is more commonly called ULTIMATE STRESS; it’s the hishes ordinate i n the curve Rupture Strength/Fracture Point – the stress at failure

=

PL E

$ = elongation P = applied force or load A = area L = original length E = modulus of elasticity + = stress . = strain b. Due to its own mass

δ

=

ρ gL2 2

=

mgL 2 E

$ = elongation % = density or unit mass of the body g = gravitational acceleration L = original length E = modulus of elasticity or Young’s modulus m = mass of the body

c. Due to changes in temperature

δ

= Lα (T f − T i )

$ = elongation ! = coefficient of linear expansion of the body L = original length Tf = final temperature Ti = initial temperature


d. Biaxial and Triaxial Deformation

µ

ε y

=−

=−

ε x

ε z ε x

& = Poisson’s ratio % = 0.25 to 0.3 for steel = 0.33 for most metals = 0.20 for concrete %min = 0 %max = 0.5 TORSIONAL SHEARING STRESS

Power delivered by a rotating shaft:

P = T ω P rpm

= 2πTN

P rpm

=

P hp

=

P hp

=

2πTN 60 2πTN

rps rpm ft − lb

550 2πTN

sec ft − lb

3300

min

Torsion – refers to twisting of solid or hollow rotating shaft.

T = torque N = revolutions/time

Solid shaft

HELICAL SPRINGS

τ

=

16T 3

πd

τ

=

τ

=

Hollow shaft

τ

16TD

=

π ( D

4

− d

4

)

' = torsional shearing stress T = torque exerted by the shaft D = outer diameter d = inner diameter Maximum twisting angle of the shaft’s fiber:

θ

=

 = angular deformation (radians) T = torque L = length of the shaft G = modulus of rigidity J = polar moment of inertia of the cross

J = J =

π ( D 4

32

πd

0.615 

 

m

Dmean d

=

Rmean

elongation,

δ

=

64 PR 3 n Gd 4

' = shearing stress $ = elongation R = mean radius d = diameter of the spring wire n = number of turns G = modulus of rigidity

 Solid shaft

− d 4 )

3

+   4m − 4 m=

πd 4 32

16 PR  4m − 1

where,

TL G

16 PR  d  + 1   πd 3  4 R 

 Hollow shaft

Gsteel = 83 GPa ; Esteel = 200 GPa


r

Mode of Interest Annually Semi-Annually Quarterly Semi-quarterly Monthly Semi-monthly Bimonthly Daily

ENGINEERING ECONOMICS 1 SIMPLE INTEREST

I = Pin

F = P (1 + in)

m 1 2 4 8 12 24 6 360

Shortcut on Effective Rate

P = principal amount F = future amount I = total interest earned i = rate of interest n = number of interest periods

Ordinary Simple Interest

n =

days

n =

360

months 12

Exact Simple Interest

n= n=

days 365 days 366

→ ordinary year → leap year

ANNUITY Note: interest must be effective rate

COMPOUND INTEREST

F = P (1 + i ) n Nominal Rate of Interest

i

=

NR

⇔ n = mN

m

Ordinary Annuity

F =

Effective Rate of Interest

ER = (1 + i )

m

−1

P =

A [(1 + i ) n − 1] i A [(1 + i ) n − 1] (1 + i ) i n

m

 NR  − 1 ER = 1 +   m  ER ≥ NR ; equal if Annual i = rate of interest per period NR = nominal rate of interest m = number of interest periods per year n = total number of interest periods N = number pf years ER = effective rate of interest

A = uniform periodic amount or annuity

Perpetuity or Perpetual Annuity

P =


A i

LINEAR / UNIFORM GRADIENT SERIES

ENGINEERING ECONOMICS 2 DEPRECIATION

P = P A P G

=

+ P G

Straight Line Method (SLM)

G  (1 + i ) n i

 

−1

i (1 + i )

G  (1 + i ) n

n

−1

−

=

AG

1  n = G −  n  i (1 + i) − 1

 

i

 n  (1 + i )  n

=

− C n n

Cm = C0 – Dm d = annual depreciation C0 = first cost Cm = book value Cn = salvage or scrap value n = life of the property Dm = total depreciation after m-years m = mth year

Perpetual Gradient

P G

C 0

Dm = md

 − n 

F G

i

d =

G i

Sinking Fund Method (SFM)

2

UNIFORM GEOMETRIC GRADIENT

d =

− C n )i n (1 + i ) − 1

(C 0

=

Dm

d [(1 + i) m

− 1]

i

Cm = C0 – Dm

 (1 + q) n (1 + i ) − n − 1 P = C   q−i  

if q * i

 (1 + q) n − (1 + i ) n  F = C   q−i  

if q * i

P =

q

=

Cn 1+ q sec ond first

P =

Cn(1 + i) n 1+ q

i = standard rate of interest Sum of the Years Digit (SYD) Method

if q = i

−1

C = initial cash flow of the geometric gradient series which occurs one period after the present q = fixed percentage or rate of increase

d m

 2(n − m + 1)  = (C 0 − C n )    n(n + 1) 

Dm

  = (C 0 − C n )  (2n − m + 1)m   n( n + 1) 

SYD =

n( n + 1) 2

Cm = C0 – Dm SYD = sum of the years digit dm = depreciation at year m


MATH and GEAS Formulas.pdf

Recommend Documents