How Do You Calculate the Inside Diameter and Mass Savings for Hollow Shafts?

[DevonZA]

Homework Statement

A solid and hollow shaft of the same material must transmit the same maximum torque. The diameter of the solid shaft is 203mm and the outside diameter of the hollow shaft is to be 216mm.
a) Determine the inside diameter of the hollow shaft. (answer = 138.7mm)
b) What percentage saving in mass would be obtained? (answer 33.46%)

Homework Equations

J=π*(D^4)/32

J=π*(D^4-d^4)/32

T/J = G * theta/l = τ/R

The Attempt at a Solution

I figured because I am only given diameters I need to use the polar second moment of area equations

For the solid shaft:
J=π*(D^4)/32
= π*(.203^4)/32
= 1.667x10^-4 m^4

For the hollow shaft (I don't think I can make the value of J the same for both the solid and hollow shaft but I don't know what else to do):
J=π*(D^4-d^4)/32
1.667x10^-4= π*(0.216^4-d^4)/32
d = 214mm

Obviously this is incorrect but I don't know where I have gone wrong

 

[haruspex]

Obviously this is incorrect but I don't know where I have gone wrong

Your algebra is all correct, even if it was guesswork. You seem to have gone wrong in the final calculation. If you cannot find the error, post all your steps there.

(You could have canceled the π and the /32 since they occur both sides of the equation.)