Equivalent torque of hollow shaft

[Deathfish]

Homework Statement

A solid circular shaft has a diameter of 200mm. What would be the dimension of a hollow
shaft which could replace it (i.e. transmit the same torque for the same maximum shear
stress), and what would be the percentage reduction in the material required? The internal
and external diameters of the hollow shaft are to be in the ratio of 4:5. (Ans: 23.84 cm,
19.07 cm; 48.9%)

Homework Equations

T/J = G(theta)/L = tao/R
J = pi(d)^4/32

The Attempt at a Solution

J = pi(0.2)^4/32 = 0.000157

pi(5d)^4/32 - pi(4d)^4/32 = 0.000157

d = 0.2m

5d = 1m (wrong)

 

[rock.freak667]

The internal and external diameters of the hollow shaft are to be in the ratio of 4:5. (Ans: 23.84 cm,
19.07 cm; 48.9%)

You are right in that you want J_solid = J_hollow

But this statement means that d/D = 4/5 or d = 0.8D (d = internal, D= external)

And J_hollow = π/32 (D⁴-d⁴)