Torsion, angle of twist, diameter, load, momentum and find G

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Discussion Overview

The discussion revolves around calculating the shear modulus (G) of a shaft given parameters such as angle of twist, diameter, uniform load, and length of the beam. Participants explore the implications of these parameters on the calculation and the necessity of integration in the process.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant expresses uncertainty about whether integration is necessary for solving the problem, given the parameters provided.
  • Another participant poses a series of questions regarding the shaft's properties, suggesting that if certain conditions are met (e.g., prismatic shaft, constant polar moment of inertia, uniform material properties, and evenly distributed torque), integration may not be required.
  • A participant indicates that their teacher advised against using a specific formula due to the presence of distributed torque, prompting further examination of the correct approach.
  • There is a discussion about the dependence of the polar moment of inertia on physical properties of the shaft, particularly questioning whether it changes with length.
  • Participants discuss how to calculate the angle of twist (θ) at the right end of the shaft based on the differential relationship between torque and shear modulus.
  • One participant mentions that they have updated their formula to include the length (L) but notes that their calculated value for G remains at 2.5 GPa.

Areas of Agreement / Disagreement

Participants generally agree on some foundational aspects of the problem, such as the conditions under which integration may not be necessary. However, there remains uncertainty regarding the polar moment of inertia and the correct application of formulas, indicating that the discussion is not fully resolved.

Contextual Notes

Participants have not definitively established the relationship between the polar moment of inertia and the length of the shaft, leaving some assumptions unverified. There is also ambiguity regarding the correct formula to use in the context of distributed torque.

roflpask
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Homework Statement


I have given angle of twist, diameter, uniform load, and length of the beam. I need to find the G.The load gives me some trouble and i don't know if i should solve it by using integration or? :)

Homework Equations

The Attempt at a Solution

 
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roflpask said:

Homework Statement


I have given angle of twist, diameter, uniform load, and length of the beam. I need to find the G.The load gives me some trouble and i don't know if i should solve it by using integration or? :)

Why do you need to integrate anything?

1. Is the shaft prismatic along its entire length?

2. Does the polar moment of inertia change w.r.t. the length of the shaft?

3. Are the material properties of the shaft the same along its length?

4. Is the torque distributed evenly w.r.t. the length of the shaft?

If you can answer "Yes" to questions 1-4, integration is not necessary. (BTW, you haven't indicated if this shaft is fixed at one end in your diagram.)

Why wouldn't you check the formula for the angular deflection of this shaft given the known quantities and see if a reasonable value of G is obtained?

Remember, you should always draw a FBD to start your analysis.
 
Thank you for your response. 1,3,4 are yes (2 dunno). I did a new drawing and wrote all the details i was given. The shaft is fixed at the end. At first i used this formula Θ=TL/GIp but teacher told me it is wrong, because i have distributed torque. After that he asked if i even know how should a stress diagram look like and i drew the same diagram as it is on the first picture.
 

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roflpask said:
Thank you for your response. 1,3,4 are yes (2 dunno).

Well, let's start with answering question 2. On what physical property of the shaft does the value of Ip depend? Is it the length? Is it the diameter? Is it something else?

Then ask yourself, does this physical property of the shaft change with length? If it doesn't, then the value of Ip stays constant.

I did a new drawing and wrote all the details i was given. The shaft is fixed at the end. At first i used this formula Θ=TL/GIp but teacher told me it is wrong, because i have distributed torque.

Let's examine your teacher's comment.

So, dθ / dx = T(x) / [G * Ip], and you want to calculate G. How do you calculate θ given dθ / dx ? Where would the value of θ = 20 mRad be obtained on this shaft?
 
The value of θ = 20 mRad would be obtained on the right end of the shaft.
 
roflpask said:
The value of θ = 20 mRad would be obtained on the right end of the shaft.

Then, which how do you find θ at the right end of the shaft given that:

dθ / dx = T(x) / [G * Ip]

and θ = 0 at the fixed end?
 
Does this look right now?
 

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You say that T(x) = M x / L, but you only use M x in your integral expression

Also, G has units, which you have neglected to show.
 
I added L to my formula but the G stays same, which is is 2,5 GPa
 

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