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Geometric Properties of Line and Area Elements Centroid Location y L = 2θ r r θ x C θ

Centroid Location y A = θr 2 r θ x C θ

r sin θ —––– θ

Ix = 14– r 4 (θ – 12– sin 2θ ) Ix = 14– r 4 (θ + 12– sin 2θ )

r sin θ —––– θ

2– 3

Circular arc segment

Area Moment of Inertia

Circular sector area y

L = π2– r

L = πr

r

C

2r — π

C

1– Ix = 16 πr4

A = 14– πr 2 4r — 3π

r

r

C

1– Iy = 16 πr4

x

4r — 3π

Quarter and semicircle arcs

y

A = –12h (a + b)

a C

h

Quarter circle area

x 2 a+ b 1– ——— 3 a+b

b

A=

π r2 —– 2

Ix = 18– π r 4

4r — 3π

r

C

h

x Iy = 18– π r 4

Semicircular area

Trapezoidal area

y

2– 5a

A = πr2 A = 23– ab

b 3– 8b

r C

C a

Ix = 14– π r 4

x

Iy = 14– π r 4

Semiparabolic area

Circular area

A = bh

y 1 ab A=— 3

b

h

3 — 10 b

C 3– 4a

x

C b

a

Exparabolic area

Ix =

1 3 — 12 bh

Iy =

1 3 — 12 hb

Rectangular area

a A=

–1 bh 2

b C

h A=

4 ab — 3

C b

2 — 5 a

Parabolic area

Triangular area

x 1– 3h

Ix =

1 3 — 36 bh

Center of Gravity and Mass Moment of Inertia of Homogeneous Solids z

z r

V = 43– π r 3

r

V = πr

G

2h

h– 2

G y

x

x

Sphere = Iyy = Izz = 25– mr 2

Ixx

1 –– 12

Ixx = Iyy =

Cylinder m(3r2 + h2) Izz = 12– mr 2 z

z

V = 13– π r 2h

V = 23– π r 3

y

h– 2

G r

h– 4

G

h

y

y r

3– r 8

x

Ixx = Iyy

Hemisphere = 0.259mr 2 Izz =

x Cone 3 2 2 –– 80 m(4r + h )

Ixx = Iyy =

2– 2 5 mr

Izz =

3 mr 2 –– 10

z z

z'

G

G r y

y

a

b x

x

Thin plate

Thin Circular disk Ixx = Iyy =

1– 2 4 mr

Izz =

1– 2 2 mr

Iz'z' =

3– 2 2 mr

Ixx =

1 –– 12

mb2 Iyy =

1 –– 12

ma2 Izz =

1 m(a2 + b2) –– 12

z z – 2 G

r G

y

x

y

– 2

x Ixx = Iyy

Thin ring = 12– mr 2 Izz = mr 2

Slender Rod

x' Ixx = Iyy =

1 –– 12

m

2

Ix'x' = Iy'y' =

y' 1– m 2 3

Iz'z' = 0