In power transmission shafts, if the polar moment of inertia of a shaft is doubled, then what is the torque required to produce the same angle of twist

Concept:

Torsion formula, \(\frac{T}{J} = \frac{\tau }{r} = \frac{{Gθ }}{L}\)

The angle of twist, \(θ = \frac{{TL}}{{GJ}}\)

Where, J = Polar moment of inertia

G = Modulus of rigidity

θ = Angle of twist in radian 

L = Length of the shaft

R = Radius of shaft

Calculation:

For the same Angle of twist, \(θ = \frac{{TL}}{{GJ}}\)

For circular section, \(J = \frac{{\pi {D^4}}}{{32}}\)

\(T \propto{J}\)

∴  \(\frac{{{T _f}}}{{{T _i}}} = \frac{2J}{J} = 2\) 

so the torque required to produce the same angle of twist will be double the original value.