Tapering Shafts:

Consider a shaft tapering from radius R1 at end to radius R2 at other end linearly as shown in Figure. Let this shaft be subjected to a torque 'T'.

Figure

Let a section at a distance x from A. The length of the strip is dx.

x → 0 to l

Radius at the section,

r = (R₁ - R₂ / l ) x

= ( R₁  - k x)

where

k =  ((R₁ - R₂ )/ l)

Consider dθ is the angle of twist on a length dx.

T/ J = G d θ/ dx

J = ( π /32 )(2π)⁴ = π r ⁴  /2

d θ=  (T /GJ ) . dx

= (T/ G) . (2/ π r ⁴)   dx

=  (2T /πG) . r ^{- 4} dx

= ( 2T /πG ). (R₁ - k _x)^{- 4} dx