Areas, Volumes, Centroids, & Moments of Inertia Plane Figures
4
Click here to load reader
description
math
Transcript of Areas, Volumes, Centroids, & Moments of Inertia Plane Figures
12/ 22/ 12 Ar eas, Volum es, Cent r oids, & M om ent s of I ner t ia Plane Figur es
1/ 4www. r oym ech. co. uk/ Usef ul_Tables/ M at hs/ M _of _I ner t ia_2. ht m l
Disclaimer: The information on this page has not been checked by an independent person. Use this information at your ow n risk.
ROYMECH
Clic k arrows to page adv erts
Home Maths Index
Properties of plane Shapes
Cx,Cy ...Centroid
The middle point of a geometric figure. The coordinates of a point in a figure, which are at the average distance from the coordinates of all points on the surface of thefigure. If the figure is 2-dimensional, the applicable term is, centre of area, if 3-dimensional the terms are, centre of volume, or centre of mass. If figure is considered to be aparticle then all of the mass or area is assumed to act at this point.
I... Moment of Inertia. /Second Moment of Area
When assessing the strength of beams to bending the "Area Moment Of Inertia" of a beams cross section indicates the beams ability to resist bending. The larger theSecond Moment of Area the less the beam will bend.
The Second Moment of Area is a geometrical property of a beam and depends on a reference axis ( which is in the plane of the area). The smallest Second Moment ofArea about any axis passes through the centroid. If the area is composed of an infinite number of small areas da the Second Moment of Area around an axis is the sum of
all these areas x (the distance of the area da from the axis) 2...This is a distance from a line
J.. Polar Moment of Inertia of a plane area
The polar moment of inertia relates to an axis which is perpendicular to the plane of an area.
If all of the area is assumed to comprise of infinitely small areas da then the polar moment of inertia is the sum of all of these areas x .r2
r = the radius of da from the perpendicular axis - for a plane area the perpendicular axis is a point
The polar moment of inertia is the sum of any two moments of inertia about axes at right angles to each other e.g.
J = Ixx + Iyy
When considering solids the Polar Moment of inertia is a measure of the resistance of a mass to angular acceleration.
Parallel Axis Theory
To determine the Second Moment of Area about an axis w hich is parallel to a centroid axis and at a distance k .
CouponDunia Free Coupons
www.coupondunia.in
India's Largest Database of Coupons Save Money With Free Coupon Codes
12/ 22/ 12 Ar eas, Volum es, Cent r oids, & M om ent s of I ner t ia Plane Figur es
2/ 4www. r oym ech. co. uk/ Usef ul_Tables/ M at hs/ M _of _I ner t ia_2. ht m l
Iw = I + A.k2
Notes
Cx , Cy = Centroid Co-ordinates.
I xx, I yy are the Moments of Area for axes through the centroid C in the direction x,y
Area = ACx
CyI xx I yy
b.h
b /2
h /2A.h 2 / 12 A.b 2 / 12
a.b.sinθ( b + a.cosθ ) / 2
a.sinθ / 2A.(a sin θ ) 2 / 12 A.(b 2 + a 2 cos 2 θ ) / 12
b.h/2
(a +b)/3
h/3A.h 2/18 A. (b 2 - a.b + a. 2 ) / 18
h.(a + b)/2 -
12/ 22/ 12 Ar eas, Volum es, Cent r oids, & M om ent s of I ner t ia Plane Figur es
3/ 4www. r oym ech. co. uk/ Usef ul_Tables/ M at hs/ M _of _I ner t ia_2. ht m l
6.a2 .tan(30o)
= 3,464 a2a /cos(30o) = 1.155 a
a
π.a2a
aA.a 2 /4 A.a 2 /4
π.(ao2 - ai
2 )
a o
a o
π.(ao4 - ai
4 )/4 π.(ao4 - ai
4 )/4
π.a2 /2
a
4.a /3.π
A.a 2 (9π 2 - 64 ) / 36.π 2
= 0,1098a 4A.a 2 /8
a 2.θ
2.a.sin θ / 3.θ
0
b.hSqrt(b2 + h2 ) /2
0/
A.b 2.h 2 / 6.( h 2 + b 2 ) A.(h 4+ b 4 ) / 12.( h 2+ b 2)
π.a.b
a
bA.b2 / 4 A.a 2 /4
12/ 22/ 12 Ar eas, Volum es, Cent r oids, & M om ent s of I ner t ia Plane Figur es
4/ 4www. r oym ech. co. uk/ Usef ul_Tables/ M at hs/ M _of _I ner t ia_2. ht m l
π.a.b /2
a
4b / 3πA.b 2 (9.π 2 - 64 ) / 36 π 2 A.a 2 /4
Links to Properties of Figures
1. Wolfram - M.of I. of common solids..High quality information2. Efunda - Common solids..Very clear comprehensive information on all important properties of solid shapes3. Chapter 5= Moment on Inertia..Download of useful review notes
T his Pa g e is b e ing d e ve lo p e d
Home Maths Index
Please Send Comments to Roy Beardmore
Last Updated 01/ 05/ 2010
Ads by Google
Shapes
3 D Shapes
Properties UK
