math guide - 34.24

1. Find all of the missing side lengths and corner angles on the two triangles below.

5

3

10°

2. Simplify the following expressions. cos θ cos θ – sin θ sinθ =

5 3

( s + 3j) ( s – 3j) ( s + 2j) 2 =

(ans.

cos θ cos θ – sin θ sinθ = cos ( θ +θ ) = cos ( 2θ )

( s + 3j) ( s – 3j) ( s + 2j) 2 = ( s2 – 9j2) ( s2 + 4js + 4j2) s4 + 4js3 + 4j2s2 – 9j2s2 – 9j24js – 9j24j2 s4 + ( 4j) s3 + ( 5) s2 + ( 36j) s + ( –36)

3. Solve the following partial fraction

4

-------------------------- = x2 + 3x + 2

Note: there was a typo here, so	1
	---------------
an acceptable answer is also. x + 0.5

(ans.

4

=

A

+

B

=

Ax + 2A + Bx + B

= ( 2A + B) + ( A + B) x

--------------------------

2

x

+ 3x + 2

x + 1

x + 2

x

2

+ 3x + 2

x

2

+ 3x + 2

A + B = 0

A = –B

4

+ –4

2A + B = 4 = – 2B + B = –B

B = –4 A = 4

x + 1 x + 2

34.3.3Geometry

•A set of the basic 2D and 3D geometric primitives are given, and the notation used is described below,

math guide - 34.25

A = contained area

P = perimeter distance

V = contained volume

S = surface area

x, y, z = centre of mass x, y, z = centroid

Ix, Iy, Iz = moment of inertia of area (or second moment of inertia)

AREA PROPERTIES:

Ix = ∫y2dA = the second moment of inertia about the y-axis

A

Iy = ∫x2dA = the second moment of inertia about the x-axis

A

Ixy	=	∫xydA =	the product of inertia
		A
JO	=	∫r2dA =	∫( x2 + y2) dA = Ix + Iy = The polar moment of inertia

AA

∫xdA

	x	------------= A	= centroid location along the x-axis
		∫	dA

A

∫ydA

	y	------------= A	= centroid location along the y-axis
		∫	dA

A

Rectangle/Square:

A = ab

P = 2a + 2b

Centroid:

b x = --2

a y = --2

math guide - 34.26

y

a

x

											b
Moment of Inertia							Moment of Inertia
(about centroid axes):							(about origin axes):
				=		ba3				=	ba3
Ix								Ix
						--------					--------
						12					3
				=		b3a				=	b3a
I								I
		y				--------			y		--------
						12					3
						= 0				=	b2a2
	I						I
			xy					xy			----------
											4

math guide - 34.27

Triangle:		y
	bh
A =	-----
			a
	2
P =			θ	h
										x
				b

Centroid:

Moment of Inertia

a + b

(about centroid axes):

=

x

3

-----------

bh

3

=

Ix

--------

h

36

y

=

--

3

=

bh

2

+ b

2

– ab)

Iy

-----( a

36

bh2

( 2a – b)

Ixy = --------

72

Moment of Inertia (about origin axes):

I		=	bh3
	x		--------
			12
			bh		2	2
Iy =			-----	( a + b – ab)
			12

Ixy =	bh2	( 2a – b)
	--------
	24

Circle:

A = π r2

P = 2π r

y

r

x

Centroid:

Moment of Inertia

Moment of Inertia

Mass Moment of Inertia

(about centroid axes):

(about origin axes):

(about centroid):

= r

π r4

Mr2

x

I

=

x

-------

Ix

=

Jz =

---------

4

2

π r4

y

= r

I

y

=

-------

Iy

=

4

= 0

Ixy

Ixy

=

math guide - 34.28

Half Circle:	y
π r2
A = -------
2
P = π r + 2r

r

x

Centroid:

Moment of Inertia

Moment of Inertia

(about centroid axes):

(about origin axes):

= r

π

8

4

4

x

=

π r

Ix

r

-- – -----

Ix

=

8

9π

-------

4r

4

8

y

= -----

π r

π r

4

Iy

3π

=

-------

I

y

=

-------

8

8

Ixy

= 0

Ixy

= 0

Quarter Circle:		y
A =	π r2
	-------
	4
P =	π r
	----- + 2r	r
	2
		x

Centroid:

Moment of Inertia

Moment of Inertia

4r

(about centroid axes):

(about origin axes):

π r

4

x

=

-----

= 0.05488r

4

Ix

=

Ix

3π

-------

16

4r

4

π r

4

y

=

-----

Iy

= 0.05488r

Iy

=

-------

3π

16

		4			r4
Ixy = –0.01647r				=
		I	xy		----
					8

math guide - 34.29

Circular Arc:			y
A =	θ r	2
	-------
	2
P = θ r + 2r			r
			x
			θ

Centroid:

Moment of Inertia

Moment of Inertia

(about centroid axes):

(about origin axes):

θ

r

4

2r sin --

( θ – sinθ

)

=

2

I

x

=

I

x

=

----

x

----------------

8

3

θ

r

4

( θ + sinθ

)

I

y

=

Iy

=

----

y

= 0

8

=

Ixy = 0

Ixy

math guide - 34.30

Ellipse:

y

r1

A = π

r1r2

r2

π

2

2

4r1∫

--

r1

+ r

2

2

P =

2

( sin θ )

dθ

x

0

1 – -------------------

a

P ≈ 2π

r

12 + r22

---------------

2

Centroid:

Moment of Inertia

Moment of Inertia

(about centroid axes): (about origin axes):

= r2

π r13r2

x

Ix =

Ix =

------------

4

= r

π r1r23

I

=

y

1

Iy =

------------

y

4

Ixy

=

Ixy

=

math guide - 34.31

Half Ellipse:								y
A =	π r1r2
	------------
	2
		π	2	2				r	1
		--	r1 + r2			2			r2
		2
P =	2r1	∫0	a		( sin θ )		dθ	+ 2r2		x
			1 – -------------------

P ≈ π	r	12 + r22
	--------------- + 2r2
		2

Centroid:

x = r2

4r1 y = -------

3π

Moment of Inertia

Moment of Inertia

(about centroid axes):

(about origin axes):

π r2r13

3

I

=

------------

Ix = 0.05488r2r1

x

16

=

0.05488r

3

=

π r23r1

I

r

I

y

2

1

y

------------

16

=

–0.01647r

2

2

=

r12r22

I

r

I

xy

1

2

xy

---------

8

math guide - 34.32

Quarter Ellipse:									y
A =	π r1r2
	------------
	4
		π	2	2					r1
	r1∫	--	r1 + r	2		2			r2
P =		2			( sin θ )		dθ	+ 2r2		x
			1 – -------------------
		0	a

P ≈	π	r	12 + r22
	--	--------------- + 2r2
	2		2

Centroid:

Moment of Inertia

Moment of Inertia

(about centroid axes):

(about origin axes):

4r2

=

=

3

=

Ix

Ix

x

-------

π r2r1

3π

4r1

3

=

Iy

=

Iy

=

π r2r1

y

-------

3π

r22r12

=

=

I

I

xy

xy

---------

8

Parabola:									y
	2
A =	3--ab								a
	b2	+ 16a2		b2			b2	+ 16a2
						4a +
P =	---------------------------		+	-----	ln	---------------------------------------			x
		2		8a			b
									b
		Centroid:				Moment of Inertia			Moment of Inertia
						(about centroid axes):			(about origin axes):

b

=

I

=

I

x

=

--

x

x

2

2a

=

I

=

=

I

y

y

y

-----

5

=

Ixy

=

Ixy

math guide - 34.33

Half Parabola:
A =	ab
	-----
	3
	b2	+ 16a2		b2	4a +		b2	+ 16a2
P =	---------------------------		+	--------	ln	---------------------------------------
		4		16a			b

y

a

x

b

Centroid:

Moment of Inertia

Moment of Inertia

(about centroid axes): (about origin axes):

3b

=

8ba3

=

2ba3

=

Ix

Ix

x

-----

-----------175

-----------7

8

19b3a

2b3a

=

=

2a

Iy

Iy

=

--------------480

-----------15

y

-----

5

b2a2

b2a2

=

=

Ixy

Ixy

----------60

----------6

• A general class of geometries are conics. This for is shown below, and can be used to represent many of the simple shapes represented by a polynomial.

math guide - 34.34

Ax2 + 2Bxy + Cy2 + 2Dx + 2Ey + F = 0
Conditions	A = B = C = 0			straight line
	B = 0, A = C			circle
	B2		– AC < 0	ellipse
	B	2	– AC = 0	parabola
	B2		– AC > 0	hyperbola

math guide - 34.35

VOLUME PROPERTIES:
			Ix	=	∫rx	2dV =			the moment of inertia about the x-axis
					V
			Iy	=	∫ry	2dV =			the moment of inertia about the y-axis
					V
			Iz	=	∫rz	2dV =			the moment of inertia about the z-axis
					V
				∫xdV
			=	V			=	centroid location along the x-axis
x
				------------
				∫	dV
				V
				∫ydV
			=	V			=	centroid location along the y-axis
y
				------------
				∫	dV
				V
				∫zdV
z			=	V			=	centroid location along the z-axis
				------------
				∫	dV

V

math guide - 34.36

Parallelepiped (box):		y
V = abc		c
		z
S = 2( ab + ac + bc)
	b	x
		a

Centroid:

			a
x		=	2--
			b
y		=	2--
			c
z		=	2--

Sphere:

4π 3 V = --3 r

S = 4π r2

Moment of Inertia (about centroid axes):

Ix =	M--------------------------( a2 + b2)
	12
Iy =	M--------------------------( a2 + c2)
	12
Iz =	M--------------------------( b2 + a2)
	12

Moment of Inertia		Mass Moment of Inertia
(about origin axes):		(about centroid):
Ix	=	Jx	=
Iy	=	Jy	=
Iz	=	Jz	=

y

r

z

x

Centroid:

Moment of Inertia

Moment of Inertia

Mass Moment of Inertia

(about centroid axes):

(about origin axes):

(about centroid):

=

2Mr2

=

=

2Mr2

= r

Ix

Ix

Jx

------------

------------

x

5

5

=

2Mr2

=

Jy

=

2Mr2

Iy

Iy

= r

------------5

------------5

y

=

2Mr2

=

=

2Mr2

Iz

Iz

Jz

z

= r

------------5

------------5

math guide - 34.37

Hemisphere:				y
V =	2	π	r	3
	--
	3			z
S =				r	x

Centroid:

Moment of Inertia

Moment of Inertia

(about centroid axes):

(about origin axes):

83

2

= r

Ix =

--------Mr

Ix =

x

320

2Mr2

3r

Iy

=

------------

Iy

=

y

= ----

5

8

83

=

2

=

Iz

Iz

z

= r

--------Mr

320

Cap of a Sphere:				y
1		2	( 3r – h)	h
V = --π h
3				z
S = 2π
	rh			r
				x

Centroid:

Moment of Inertia

Moment of Inertia

(about centroid axes):

(about origin axes):

=

Ix

=

= r

Ix

x

=

Iy

=

=

Iy

y

=

Iz

=

z

= r

Iz

math guide - 34.38

Cylinder:	y	r
V = hπ r2		h
S = 2π rh + 2π r2		z
		x

Centroid:

Moment of Inertia

Moment of Inertia

(about centroid axis):

(about origin axis):

=

h2

r2

h2

r2

x

r

Ix

=

M

Ix

=

M

-----

+ ----

----

+ ----

12

4

3

4

h

Mr2

y

=

Iy

=

---------

Iy

=

--

2

2

=

h2

r2

=

h2

r2

I

M

I

M

z = r

-----

+ ----

----

+ ----

z

12

4

z

3

4

Mass Moment of Inertia (about centroid):

	Jx	=	M-----------------------------( 3r2 + h2)
			12
	Jy	=	Mr2
			---------2
	Jz	=	M-----------------------------( 3r2 + h2)
			12

Cone:						y
V =	1	π	r	2	h
	--
	3
S = π r r2 + h2						z	h
							r
							x

Centroid:

Moment of Inertia

Moment of Inertia

(about centroid axes):

(about origin axes):

=

3h3

3r2

x

r

I

=

M

I

=

x

--------

+ -------

x

80

20

h

=

3Mr2

=

=

Iy

Iy

y

--

------------

4

10

=

3h3

3r2

=

I

M

I

z

= r

--------

+ -------

z

80

20

z

Tetrahedron:

1

V = -3-Ah

math guide - 34.39

y	z
h
A	x

Centroid:

Moment of Inertia

Moment of Inertia

(about centroid axes):

(about origin axes):

=

Ix

=

=

Ix

x

=

Iy

=

h

Iy

y

=

4--

=

Iz

=

z

=

Iz

Torus:										y
1		2								r2
	π		( r		+ r		) ( r		– r )	2
V = --				1		2		2
4									1	r
											1
										z
S = π 2( r22 – r12)											x

Centroid:

Moment of Inertia

Moment of Inertia

(about centroid axes):

(about origin axes):

=

Ix

=

= r2

Ix

x

r2

– r1

=

I

=

I

y

y

y

= --------------

2

=

Iz

=

z

= r2

Iz

math guide - 34.40

Ellipsoid:								y
		4	π r					r2	r3
V	=				r		r
		--		1		2		3
		3							r1
								z
S =									x

	Centroid:				Moment of Inertia					Moment of Inertia
					(about centroid axes):					(about origin axes):
									=	Ix	=
				= r1		Ix
	x
									=	Iy	=
				= r2		Iy
	y
									=	Iz	=
	z			= r3			Iz

Paraboloid:					y
V =	1	π r	2	h
	--				h
	2
					z
S =					r	x

Centroid:

Moment of Inertia

Moment of Inertia

(about centroid axes):

(about origin axes):

=

Ix

=

= r

Ix

x

=

Iy

=

=

Iy

y

=

Iz

=

z

= r

Iz