Polar moments and products of inertia

(Including accompanying parallel-axis theorems)

Polar moments of inertia

The polar moment of inertia with respect to an axis through O perpendicular to the plane of the figure is defined by the integral:

To illustrate this integral, let's look at an arbitrary figure with a differential area dA:

We can see from the above figure that the integral of polar moment of inertia is similar to those for moments of inertia I_x and I_y. In fact:

This is a important relationship because it shows that 'the polar moment of inertia with respect to an axis perpendicular to the plane of the figure at any point O is equal to the sum of the moments of inertia with respect to any two perpendicular axes x and y passing through that same point and lying in the plane of the figure' (Gere 2004, p.841).

Parallel-axis theorem for polar moments of inertia

The above theory can be furthered to include the parallel-axis theorem for polar moments of inertia:

Figure 1

Let's designate the polar moments of inertia at points O and C by I_po and I_pc, respectively. Then, by using the equation from polar moment of inertia derived above, we can write:

I_po = I_x + I_y     and     I_pc = I_xc + I_yc

Now we use the parallel-axis theorem to get:

I_x + I_y = I_xc + I_yc + A(a² + b²)

I_po = I_pc + A(a² + b²)

In fact a² + b² is the squared distance from point O to C.

Products of inertia

Integral for the product of inertia with respect to the x and y axes is given by:

The result may be positive, negative or zero depending on where the x and y axes are located. Also, if an axis acts through the line of symmetry of the object, its product of inertia will be zero.

Parallel-axis theorem for products of inertia

If we refer back to figure 1 (above), we can derive the following expression for the parallel-axis theorem for products of inertia:

References:

• Gere J.M. 2004, Mechanics of Materials, 6^th edn, Thomson, USA