Engineering Statics

F-29

(a)Determine by direct integration the polar moment of inertia of the annular area shown with respect to point O. (b)Using the result of part a, determine the moment of inertia of the given area with respect to the x axis.
The expression for the infinitessimal polar moment of inertia is given in terms of r and dA.
The infinitessimal area is given as a infinitely thin ring of radius r and thickness dr.
The expression for the polar moment of inertia is written as an integral. The expression for dA is substituted into the equation and the limits of integration go from the radius of the inner circle to the radius of the outer circle.
The integrand is simplified.
The integration is performed and limits are substituted into the expression.
The polar moment of inertia is the sum of the moments of inertia about two orthogonal axis. In this case, because of symmetry, the moment of inertia about the y and x axis are equal.
Dividing the polar moment of inertia by 2 gives the desired answer.