November 29, 2003
A: FROM MENTOR ANDREA GUNTHER IN WA
Your questions show you want to understand why all of
these formulas work and not just memorize them.
Understanding the how and why of science will help you
remember the concepts and understand them more deeply.
I will try my best to answer them and please ask for
more explanation if this is not enough.
1) Moment of Inertia: The property of an object
associated with its resistance to rotation. It
depends on the objects mass and the distribution of
mass with respect to the axis of rotation.
The distribution of the mass with respect to the
center is what gives the formula. The further the
mass is from the center, the greater the resistance to
rotation (or bending). A hollow pipe has a cross
section of a circle. The center of the circle is the
center of mass. The radius of the circle is how far
the mass is from the center. If that same mass is
squished towards the middle like a solid rod, the same
amount of material would have less resistance to
bending. The calculus is : Integration of Ad^2 over
the entire area. (A = area, d= distance to the center
of the cross section)

2) Resistance to bending: Resistance to bending has
two parts. One is the Moment of Inertia or shape of
the cross section, and Two is the stiffness of the
material which is its resistance to stretching (it is
called Young’s modulus or E).
Bending: Bending can be thought of as two forces
acting in opposite direction with a distance between
them. If you bend an item, one side is being
compressed and the other side is being stretched. If
you break a wood pencil by bending it, it fails on the
tension side by cracking.

a. Moment of Inertia is important because the further
the material is away from the center the greater
bending it can react. Imagine the cross section
divided into tiny bits. Each bit is reacting some
force. The bit will either be compressed or
stretched. The distance of the bit to the center is
like its lever arm. When you pry on something the
further you are from the fulcrum or the longer the
lever arm, the more force you can exert. Therefore
the more bits that are away from the center, the more
bending it can react. This is why an I beam (an I
beam is a beam shaped like a capital I) can react a
lot of bending. The top and bottom of the I shape are
far from the center of the beam.
b. Stiffness of material: The material stiffness is
defined as the amount of length change versus the
applied load. Material that has very little
stiffness, like a rubber band, will have a lot of
length change for an applied load. Something stiffer,
like a steel rod will not stretch as much with the
same applied load. If I want to stretch each item 1
inch, it would take an incredible amount of load to
stretch the steel and not very much to stretch the
rubber band. Because bending is just putting some
bits is compression and some bits in tension, not as
much bending is required to deflect a rubber item as
it would a steel item.

3) Hydrostatic pressure and soil: Without seeing how
your book and teacher define liquid and solid density
I will be taking a guess at what your question is.
If you have a scoop of dirt, there are three distinct
things. There is water, dirt and voids(air). Density
is mass per volume.

Liquid Density: I think you may be referring to the
wet density. This is where all the voids are filled
with water. In this case, you have no voids and have
only water and dirt. This may also refer to the
density of the liquid by itself.

Solid Density: I think this refers to the density of
the dirt alone.