If you take the limiting case of R=0 you get the thin rod expression, and if you take the case where L=0 you get the thin disk expression.
Moment of inertia equation plus#
This form can be seen to be plausible it you note that it is the sum of the expressions for a thin disk about a diameter plus the expression for a thin rod about its end. Now expressing the mass element dm in terms of z, we can integrate over the length of the cylinder. For any given disk at distance z from the x axis, using the parallel axis theorem gives the moment of inertia about the x axis. This involves an integral from z=0 to z=L.
Moment of inertia equation full#
Obtaining the moment of inertia of the full cylinder about a diameter at its end involves summing over an infinite number of thin disks at different distances from that axis. The approach involves finding an expression for a thin disk at distance z from the axis and summing over all such disks. The development of the expression for the moment of inertia of a cylinder about a diameter at its end (the x-axis in the diagram) makes use of both the parallel axis theorem and the perpendicular axis theorem. Moment of Inertia: Cylinder About Perpendicular Axis The only difference from the solid cylinder is that the integration takes place from the inner radius a to the outer radius b: Show development of thin shell integral The process involves adding up the moments of infinitesmally thin cylindrical shells. The expression for the moment of inertia of a hollow cylinder or hoop of finite thickness is obtained by the same process as that for a solid cylinder. Substituting gives a polynomial form integral:
The mass element can be expressed in terms of an infinitesmal radial thickness dr by Using the general definition for moment of inertia: Moment of Inertia: CylinderThe expression for the moment of inertia of a solid cylinder can be built up from the moment of inertia of thin cylindrical shells.
Show development of expressions Hollow cylinder case The moments of inertia for the limiting geometries with this mass are: I thin disk diameter = kg m 2 Length L = m,the moments of inertia of a cylinder about other axes are shown. Will have a moment of inertia about its central axis: I central axis = kg m 2 The polar moment of inertia describes the distribution of the area of a body with respect to a point in the plane of the body. (10.5.1) J O A r 2 d A, where r is the distance from the reference point to a differential element of area d A. Parallel Axis Theorem Moment of Inertia: Cylinder The polar moment of inertia is defined by the integral quantity.
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