The ratio of polar moment of inertia to the planar moment of inertia for a solid shaft of diameter D is given by ____________.

Explanation:

Planar moment of inertia:

Planar moment of inertia describes how an area is distributed relative to a reference axis. This is important because it specifies the area’s resistance to bending.  Planar moment of inertia is denoted with the letter I.

Polar Moment of Inertia:

Polar moment of inertia is analogous to the planar moment of inertia but is applicable to a cylindrical object and describes its resistance to torsion (twisting due to an applied torque). Polar moment of inertia is sometimes denoted with the letter J or I_P.

The relation can be derived from Perpendicular axis theorem.

Perpendicular axis theorem: 

This theorem states that the moment of inertia of a plane lamina about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two perpendicular axes concurrent with perpendicular axis and lying in the plane of the body.

It can be expressed as Izz = Ixx + Iyy

Here Ixx, Iyy and Izz are the moment of inertia of an object along X, Y and Z axis respectively out of which

Ixx and Iyy  are the planar moments of inertia and Izz is Polar moment of inertia for the given figure.

For solid shaft cross-section:

Planar moment of inertia

$I=\frac{\pi D^4}{64}{m}^4$

Polar Moment of Inertia

$I_P=\frac{\pi D^4}{64}+\frac{\pi D^4}{64}$

$I_P=\frac{\pi D^4}{32}{m}^4$

Hence, $\frac{I_p}{I}=2$

Additional Information

The equation for polar moment of inertia is essentially the same as that for the planar moment of inertia, but in the case of polar moment, distance is measured to an axis parallel to the area’s cross-section, instead of I, but its units are the same as those for the planar moment of inertia i.e. m⁴.