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Converted document INERTIA PROPERTIES EXAMPLES

INERTIA PROPERTIES EXAMPLES

This example demonstrates how to compute the moment of inertia of a uniform cylindrical rod about its axis. An example of how moments of inertia are used in solving engineering problems is given in the Deriving Equations of Motion section and Angular Momentum section.

Consider a rod with mass density ρ, length L, and radius R. The equation for the moment of inertia is

(1) I = ⌠⌡_Vρr⋅rdV

The volume can be expressed as V = πR²L, and so dV = 2πLrdr, where dr is an infinitesimal section of R. Equation 1↑ may then be rewritten as

(2) I = ^R⌠⌡₀ρr²(2πLrdr)

For a uniform rod, the mass density ρ is constant, so it can be taken out of the integral and the result is

(3) I = 2πρL^R⌠⌡₀r³dr = (1)/(2)πρLR⁴

In many cases for a uniform rod, the total mass m is given instead of the mass density. For a uniform rod, the mass is m = ρV = ρπR²L. Using the mass instead of the mass density, the moment of inertia about the cylinder’s axis is

(4) I = (1)/(2)mR²