Understanding Homework Equation: Lφ=rθ

[Suraj M]

Homework Statement

Homework Equations

$L\phi = r \theta$

The Attempt at a Solution

Firstly I'm not sure what $\phi$ and $\theta$ mean in the equation,
If I were to attempt I'd want to equate the lengths
$$ \frac{2r\theta_1}{\phi} = \frac{r\theta_2}{\phi} $$
I felt $\theta_2$ is zero and $\theta_1 = \phi in the question$
Definitely wrong
Could you help?

 

[stockzahn]

The cylinders are made of the same material, they are of the same length and the torque is constant vs. the whole length - their torsional stiffness and the angle only depend on the geometry.

First of all: Can you eliminate two answers without any calculations?
Secondly: Do you know what geometrical dependent property defines the torsional stiffness of a body?

 

[Suraj M]

C and D are gone since they're more than the original angle
Geometrical dependent property?[emoji20][emoji17] sorry but no
Where can o find out more
Our teacher didn't even teach us this topic but there are questions on it [emoji53]

 

[stockzahn]

C and D are gone since they're more than the original angle
Geometrical dependent property?[emoji20][emoji17] sorry but no
Where can o find out more
Our teacher didn't even teach us this topic but there are questions on it [emoji53]

The torsional stiffness depends on the geometry of the body - it seems to be quite obvious, that the cylinder with the larger diameter is more stiff, than the cylinder with the smaller diameter. But how much? You need the look up the area moment of inertia. This property is a measure for the resistance of the body against deformation. In your case, a body applied to a torque, you need to find out the polar area moment of inertia of a rigid cylinder.

Comparing the polar area moments of inertia of two cylinders with r and 2⋅r (by dividing one by the other) will give the answer to your question.

EDIT: Your thougt regarding answers C and D is correct.

 

[Nidum]

Here is a simple tutorial on torsion of shafts which will help you understand the background theory to your problem :

http://www.slideshare.net/osmak93/mm2105a

 

[haruspex]

If you work out what the a and b choices translate into in terms the ratio between the twist in one cylinder and the twist in the other, it's kind of obvious which one to choose, even without understanding any of the theory.

 

[Suraj M]

Here is a simple tutorial on torsion of shafts which will help you understand the background theory to your problem :

http://www.slideshare.net/osmak93/mm2105a

I'm really sorry but I didn't quite follow after they derived the basic formula relating length radius and angle

 

[Suraj M]

If you work out what the a and b choices translate into in terms the ratio between the twist in one cylinder and the twist in the other, it's kind of obvious which one to choose, even without understanding any of the theory.

How could I decide between A and B without solving?

 

[Suraj M]

Oh wait
Is it like this?
$$ \frac{\phi - \theta}{\theta-0} = (\frac{r}{2r})^4 $$??
Then I get option B
Right?

 

[stockzahn]

Oh wait
Is it like this?
$$ \frac{\phi - \theta}{\theta-0} = (\frac{r}{2r})^4 $$??
Then I get option B
Right?

Where did you get the formula you use (and you showed in your first post) from? And what does θ stand for in this equation? It doesn't seem very common to me, I would try to solve the task in a different way - but I don't have to deal with strength of materials on a regular basis, so maybe my thought is not correct.

 

[haruspex]

Oh wait
Is it like this?
$$ \frac{\phi - \theta}{\theta-0} = (\frac{r}{2r})^4 $$??
Then I get option B
Right?

Yes.

 

[Suraj M]

Where did you get the formula you use (and you showed in your first post) from? And what does θ stand for in this equation? It doesn't seem very common to me, I would try to solve the task in a different way - but I don't have to deal with strength of materials on a regular basis, so maybe my thought is not correct.

The first formula in the original post is the same as the one in the link Nidum provided
Except the change in symbols
That theta isn't the one I used here in post #9
I'm sorry for that
I took theta to be the angle of twist at the junction
And then I thought that the angle would be inversely proportional to the polar moment of inertia, which is actually a function of $r^4$
So I equated that
As I've shown
I don't know if it's right
I just used some basic logic to link the difference in the angle at one end and the resulting twist angle at the other end to radius
Could you direct me to a more refined way?

 

[Suraj M]

Yes.

And this-?

How could I decide between A and B without solving?

 

[haruspex]

And this-?

Having got it down to a choice of A 15:1, and B 16:1, and knowing the radius ratio is 2:1, the twist ratio of 16:1 looks far more likely.

 

[Suraj M]

Having got it down to a choice of A 15:1, and B 16:1, and knowing the radius ratio is 2:1, the twist ratio of 16:1 looks far more likely.

Ah! Okey
Thanks a ton EVERYONE
Really appreciate it [emoji4]