|
1.Rotational Inertia of Geometrical Bodies
Click For PDF Version
(a) Annular cylinder about its central axis Let 
We will calculate expression for the rotational inertia by integrating with variable r, the radial distance measured from the axis. Mass of annular cylinder is given by the integral
Its rotational inertia is given by the integral
(b) Solid cylinder (or ring) about central axis *Move the Mouse over the image for an Animated View
Let the radius of the cylinder be R and its mass M.
We can obtain its rotational inertia I from the formula for the
rotational inertia of an annular cylinder by substituting
We have
(c) Solid disk of width 
Let R be the radius,
The moment of inertia can be found by integrating
As
Therefore,
But the mass of the disk is
Thus the moment of inertia of a thin disk of mass M is
We will use the parallel axis theorem for finding the rotational inertia of a thin disk about an axis parallel to the vertical axis passing through its centre.
This gives
Using the expression of rotational inertia of a thin disk, we have
(d) Cylinder about axis through its CM We will use this result for calculating the rotational inertia of a solid cylinder of length L, radius R, and mass M about a vertical axis passing through its centre of mass. *Move the Mouse over the image for an Animated View
Mass of the cylinder M is
We thus find that the rotational inertia of a cylinder about axis as shown in the figure is
(e) Thin rod about an axis through its centre *Move the Mouse over the image for an Animated View
Rotational inertia of a thin rod of length L and mass M about an axis passing through its centre can be obtained from the above result by putting in it R =0. We get
(f) Thin rod about axis at one of its ends *Move the Mouse over the image for an Animated View
By applying parallel axis theorem and using the expression of rotational inertia of a thin rod about axis through its CM, we get
(g) Thin spherical shell about any diameter Let radius of the shell be r, its thickness
Mass of a spherical shell of radius r, thickness
Using this expression for M, the rotational inertia of a thin spherical shell of radius r can be expressed as
(h) Rotational inertia of a solid sphere With this result we obtain next the rotational inertia of a solid sphere of radius R about any diameter. *Move the Mouse over the image for an Animated View
Integrating the thin spherical shell expression with respect to r from 0 to R, we get
Mass of a homogeneous sphere of radius R and density
We thus find
(i) Rotational Inertia of Thin Slab*Move the Mouse over the image for an Animated View 
Let the length of the slab be a, its width be b, its thickness
be
Mass of the slab is
We thus find
|
be the outer
radius of the annular cylinder and
be its inner radius, and
l be its length. Let
be its density.
thickness and
be the density of the
disk. For calculating the rotational inertia about the axis as shown in the
figure we choose angular variable
measured from the
vertical direction, and consider an infinitesimal box of length dx,
height dy and width 
and
be its density. Using
spherical polar coordinates and measuring distance from the polar axis, we have
and density
is
is
and its density be
. Its rotational inertia
about an axis perpendicular to its plane and passing through its centre of mass
can be calculated by integrating the following expression;
