What is the Angle of Twist for a Hollow Tube with a Cut Along its Length?

[Corsan]

Homework Statement

I have been tasked with finding the angle of twist of the following component:
Hollow tube 40mm diameter, 2mm wall thickness, with a 2mm cut along its length (see image), x=2mm.

Homework Equations

Please see attachment for equations.

The Attempt at a Solution

Please see attachment for my attempt.
I calculated this to be 1.70x10^-8 radians/metre.
This seems very small (incorrect?).
Can anyone spot any obvious errors with the equations/workings out?
Thanks in advance

 

[nvn]

Corsan: You need to convert all values to consistent units. List the units with each quantity in your calculations, and ensure the units cancel out to produce the correct units for the answer. Is 40 mm the tube inside diameter, outside diameter, or mean diameter? You want to ensure you are using the tube mean circumference. Also, do not subtract the saw-cut gap from the tube radius; subtract the saw-cut gap from the tube mean circumference. Try again.

 

[p21bass]

In addition to what nvn posted, there's a couple of other things that stick out.

First, your answer shouldn't be in rad/m. It would just be in radians (or degrees). The angle is consistent along the entire length of the tube. It's the arc length that will change.

Second, you may find it easier to calculate your polar moment of inertia separately, and then plug it into your equation. That way, you can limit the amount of units you have to deal with at anyone time. It's fine to incorporate it all into one equation - I just (personally) find that it sometimes gets a little messy with that many numbers and units.

 

[nvn]

No tube length is given. Therefore, Corsan would need to report radians/metre (if no tube length is given). The twist angle changes constantly (linearly) along the tube length.

 

[p21bass]

Right - gamma is the constant angle, but theta (what's actually being solved for) changes. My bad.

In the case of no tube length given, then the solution for theta must include L (as an unknown) in its answer. Otherwise, rad/meter is a ratio, and not really an angle. So, including the unknown L as part of the answer indicates that the length (in meters) "cancels out" the meter in the denominator.