:

I m AB o m :

I = m 2 (1) m . - Definition : The moment of intertia of a particle of mass m about a line or axis AB is defined as I = where is the distance from the mass to the line. Note that the moment of inertia is positive amount or zero. Z m (x, y, z)

o Z y

x

x y

m (x, y, z) o x y z oz, oy, ox Izz, Iyy, Ixx . - Suppose that a particle of mass m at point (x,y,z) The moment of intertia about the axis ox , oy , oz are Ixx = m (y2 + z2) , Iyy = m (x2 + z2) Izz = m (x2 + y2 ) (2.a , b, c)

o Io :- The moment of inertia of a particle about the origion point O isIo = m (x2 + y2 + z2) (3) y o z ( (x = o :- The moment of inertia of a particle about planes x = o , y = o , z = o respectively defined from:Ix=o = m x2 (4) y = o z = o : Iy=o = m y2 , Iz=o = m z2 (5) : - The moment of inertia of a rigid body d m (x, y, z) o x y z oz , oy , ox : Ixx = (y2 + z2) d m Iyy = (z2 + x2) d m Izz = (x2 + y2) d m (6.a, b, c)

o : Io = (x2 + y2 + z2) d m (7)

z = o , y = o , x = o :Ix=o = x2 d m , Iy=o = y2 d m , Iz=o = z2 d m (8): - Perpendicular Axis theorem : Izz = (x2 + y2 ) d m = x2 d m + y2 d m = Ixx + Iyy .

: - Radius of Gyration k m :

I = m k2 (9) I (9). : - Parallel Axis Theorem " " :

o x` d o x , o` x`. : 1- : - 2a m :

- - - a/2

- Find the moment of inertia of a uniform rod of length 2a about an axis perpendicular to it and passing through (a) the center of mass (b) an end (c) a point at distance from an end .

x

o x dx a a

m 2a . oy o :

dm dx x oy . ( dm = ( d x :

: (i)- k oy .

(ii)- ( ox) .

(iii)- . y` y

a o

x

2- : - 2a , 2b .

- Find the (a) moment of inertia and (b) radius of gyration of a rectangular plate with sides 2a and 2b about a side and ends . A B C D m 2a 2b . dm x oy :

D y c

b

b

A a a B oy. ox :

: 1- AB .

2- AD :

3- oz o : - Find the moment of inertia of a rectangular plate with sides 2a and 2b about an axis perpendicular to the plate and passing through a vertex.

- Ez` E AD :

z` D z y c

E

x

A a a B

3- : -

- Find the moment of inertia of a ring about a diameter . xoy m a o oz .

z

o y

x dm

: Ixx = Iyy( Izz = Ixx + Iyy

( Ixx = Iyy = m a2 ma2 4- : - - Find the moment of inertia of a circular disk about a diameter . z

y x

m o a xoy o oz

r dm : d m = ( 2 ( r d r

( ( ) dr

m = ( ( a2 ( Izz = m a2

(5) : - a h m

- Find the moment of inertia of a hollow cylinder of radius a , height h and mass m about the axis of the cylinder . z

dm oz :

m a .

6- : - a h m

- find the moment of inertia of a solid circular cylinder of radius a , height h and mass m about (a) the axis of the cylinder (b) an axis parallel to the axis of the cylinder and distant b from its center . oz :

m a . - Use the parallel axis theorem to find the moment of inertia of a solid circular cylinder about a line on the surface of the cylinder and parallel to the axis of the cylinder .

(7) : -

- Find the moment of inertia of a hollow sphere about a diameter . o Io = ma2 a, m : ( )Ixx + Iyy + Izz = 2Io,

Ixx = Iyy = Izz = I

.

8- : - - Find the moment of inertia of a solid sphere about a diameter.

r dr

:

(9) : - h a .

- Find the moment of inertia of a hollow cone of height h and radius a about its axis . y h ds

o x y a x ox

dm () d m = ( 2( y ds

s, y

m

10- : - h a

- Find the moment of inertia of a right circular cone of height h and radius a about its axis . y

h

o x y a x

ox

y, x ( )

o

x

b

b

o

a

dr

r

o

a

o

dr

o

r

EMBED Equation.3

PAGE 12

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