This document discusses various properties of surfaces and solids including: 1) Centroid or center of gravity is the point where the entire weight of a body is assumed to act vertically downward. 2) Pappus and Guldinus theorems relate the surface area and volume of revolved surfaces and solids to the generating curve and plane. 3) Parallel axis theorem states the moment of inertia about any axis equals the moment about a parallel centroidal axis plus the product of the area and square of the distance between axes.

1. UNIT III - PROPERTIES OF
SURFACES AND SOLIDS
Mr.B.K.Parrthipan, M.E., M.B.A., (Ph.D).,
Assistant Professor / Mechanical Engineering,
Kamaraj College of Engineering and Technology.
ENGINEERING MECHANICS

2. Centroid or Centre of Gravity
Centroid or Centre of gravity of a particularly
configured body is a point through which its weight
is assumed to be acting vertically downwards.

3. Theorems of Pappus and Guldinus
Pappus (Greek Scientist) and Guldinus ( Swiss
mathematician) developed two theorems to
determine surface area and volume of solid bodies
knowing of a curve and an area respectively.

4. Pappus and Guldinus Theorem 1
It states that “ the area of a surface of revolution is
the product of the length of the generating curve
and the distance travelled by the centroid of the
curve, while the surface is generated.

5. Pappus and Guldinus Theorem 2
It states that “ the volume of a solid body of
revolution is equal to the product of the area of the
generating plane and the distance travelled by the
centroid of the generating plane while the body is
being generated.

6. Parallel axis theorem
It states that “ the moment of inertia of a lamina
about any axis in the plane of lamina is equal to the
sum of the moment of inertia about a parallel
centroidal axis in the plane of lamina and the
product of the area of the lamina and square of the
distance between the axes. “
IAB=IXX+Ah2

7. Perpendicular Axis Theorem
It states that “ If IOX and IOY be the moment of
inertia of a lamina about two mutually
perpendicular axes OX and OY in the plane of the
lamina and IOZ be the moment of inertia of the
lamina about an axis normal to the lamina and
passing through the point of intersection of the axes
OX and OY, then
IOZ= IOX+IOY

8. Principle Moment of Inertia
The perpendicular axes about which product of
inertia is zero are called “Principle axes” and the
moment of inertia with respect to these axes are
called as “Principle Moments of Inertia”

9. Polar Moment of Inertia
The polar moment of inertia is equal to the sum of
the area moments of inertia about any two mutually
perpendicular axes in its plane and intersecting on
the polar axis.

10. Radius of Gyration
Radius of gyration about an axis is defined as the
distance from that axis at which all the elemental
parts of the lamina would have to be placed such
that the moment of inertia about the axis is same.
I=Ar2
Where r – radius of gyration

11. Centroid

12. Centroid

13. Moment of Inertia

14. Moment of Inertia