In the context of the torsion equation, the polar moment of inertia (J) for a solid circular shaft of radius R is given by which of the following formulas? 

CONCEPT:

• Polar Moment of Inertia is a measure of an object’s capacity to oppose or resist torsion when some amount of torque is applied to it on a specified axis.
• Polar Moment of Inertia also known as the second polar moment of area is a quantity used to describe resistance to torsional deformation.
• It is denoted as Iz or J.
• The moment of inertia for a solid circular beam with diameter d is

  \(⇒ I_{xx}=I_{yy}= \frac{{\pi \left( {d^4} \right)}}{{64}}\) 

Where Ixx and Iyy are moments of inertial w.r.t. xx axis and yaxis respectively.

We know that

Polar moment of inertia. For objects that have rotational symmetry such as a solid cylinder the equation can be simplified to

⇒ Jz = Ixx + Iyy = 2Ixx = 2Iyy

\(\Rightarrow J_z = 2 \times \frac{{{\rm{\pi }}\left({{\rm{d}}^4} \right)}}{{64}} = \frac{{{\rm{\pi }}\left({{\rm{d}^4}} \right)}}{{32}}\)

∵ d = 2R

∴ J = \(\rm \frac{\pi R^4}{2}\)