Unformatted text preview: ht radius = r f(x) x =h The cylinder is known as a solid of rotation . The technique used in example 8
solid
to find the volume of a cylinder cannot be extended to other solids of rotation.
solids
Because we cannot get an accurate measure of the element of volume dV.
element volume
We need an accurate measure of dV.
In the case of a cylinder we may take an element of volume dV to be a solid disc
element
of radius f(x) and thickness dx. Then :
dV = π {f(x)}2.dx = πr2.dx h h h 0 0 0 volume of a cylinder = ∫dV = ∫ π {f(x)}2.dx = ∫ πr2.dx = [ πr2x] = πr2h
This technique may now be applied to other solids of rotation.
solids
h h volume of cone = ∫dV = ∫ π {f(x)}2.dx = ∫ π(rx/ h)2.dx
0 23
= [ π(r x / 3h2)] h 0 192 0 = 1 3 πr2h
/ The physical entities like position , distance , speed , acceleration , area and
position distance speed acceleration area
volume are easy to visualize. Let us now look at an application of integration where
the physical entity being measured is not so obvious. M oment of I ner tia
In linear motion t he distribution of the mass (density) of the rigid body
l inear
does not matter. We may take the center of mass of the rigid body and treat
c enter
it as concentrated mass or point mass . So we have only one position vector for
point
orientation
the rigid body. Hence the orientation of the rigid body does not affect the
orienta
equation of energy of linear motion. In rotational motion the orientation of
rotational
orientation
the body does affect the equation of energy of rotational motion. orientation = distance of the individual elements of mass of the rigid body
elements
from a particular point or axis.
elements of mass = similar segments or crosssections of the mass, more than
just a particle.
Contiguous elements of mass when summed up continuously or integrated form
elements
the entire mass of the rigid body . To keep our calculations simple at the high
school level, we assume the mass of the rigid body to be of unifor m density .
unifor
u niform
Hence they are of equal m...
View
Full Document
 Fall '09
 TAMERDOğAN
 Limit, Δx

Click to edit the document details