1.Rotational Inertia of Geometrical Bodies
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(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 |