Mechanics of Materials - Torsion

[Femme_physics]

Homework Statement

http://img840.imageshack.us/img840/736/mmaterials.jpg

Homework Equations

http://img18.imageshack.us/img18/8739/releq.jpg

The Attempt at a Solution

http://img845.imageshack.us/img845/7017/pitul.jpg

The answers turn up (a) 56.6 MPa and (b) 36.6 MPa so I'm way off. Where did I fall short?

 

[NascentOxygen]

F_AB is the force acting at the radius of the shaft. You used its torque.

0.15 is in mm, SI formulae require metres.

Fix these and you'll get 56.6 MPa

 

[PhanthomJay]

Your first relevant equation is not correct and your second seems to be an allowable shear stress which is not applicable in this problem.
For a circular member subject to torsion from 'twisting' moments about the long axis, you should look up the formula for max shear torsional stress. The formula you noted applies for average transverse shear streses parallel to the cross sectional area.

 

[Femme_physics]

FAB is the force acting at the radius of the shaft. You used its torque.

Yes, and I believed I used the right value.

0.15 is in mm, SI formulae require metres.

Fix these and you'll get 56.6 MPa

I fixed it to "M" but I still get the same numbers just the decimal point shifting.

Your first relevant equation is not correct and your second seems to be an allowable shear stress which is not applicable in this problem.

Not correct by not being relevant or not correct by not correct? My teacher gave us those formulas.

For a circular member subject to torsion from 'twisting' moments about the long axis, you should look up the formula for max shear torsional stress. The formula you noted applies for average transverse shear streses parallel to the cross sectional area.

This formula?

http://img42.imageshack.us/img42/2406/etaz.jpg

 

[PhanthomJay]

Not correct by not being relevant or not correct by not correct? My teacher gave us those formulas.

not relevant

This formula?

http://img42.imageshack.us/img42/2406/etaz.jpg

Yes...where Io is the polar moment of inertia about the longitudinal axis of the shaft.

 

[NascentOxygen]

Yes, and I believed I used the right value.

You used the right value of torque, but you didn't convert it to its tangential force equivalent at the radial distance from the shaft's axis. The formula you have been given requires the tangential force applied at the shaft surface. Force = torque / radius

I fixed it to "M" but I still get the same numbers just the decimal point shifting.

I just rechecked, and I get book's answer.

Not correct by not being relevant or not correct by not correct? My teacher gave us those formulas.

I verified your teacher's formula before I replied the first time, to confirm it derives from the general formula. It's correct.

 

[I like Serena]

Erm...
$${T/r \over \pi r^2} = {300 Nm / 0.015 m \over \pi (0.015 m)^2} = 28.3 MPa \ne 56.6 MPa$$

 

[NascentOxygen]

Erm...
$${T/r \over \pi r^2} = {300 Nm / 0.015 m \over \pi (0.015 m)^2} = 28.3 MPa \ne 56.6 MPa$$

Ah, there's a 2 that has gone missing somewhere. The polar moment of a round shaft is Pi/2 times r^4. The teacher's formula seems to be out by a factor of two.

I guess this means no one (except me) will have got the right answer?

 

[I like Serena]

Ah, there's a 2 that has gone missing somewhere. The polar moment of a round shaft is Pi/2 times r^4. The teacher's formula seems to be out by a factor of two.

I guess this means no one (except me) will have got the right answer?

I believe PhantomJay had it right when he said Fp needed another formula that included a polar moment, which she just gave.

The first formula in the OP is apparently for the shear tension, not the shear torsion.

 

[NascentOxygen]

Determine maximum shear stress on shaft, $\tau _{max}$, from the general formula:

$\frac T J = \frac {\tau_{max}} {r} \; \;\; where\;\; for\;\; a\;\; solid\;\; cylindrical\; shaft\;\;J = \frac {\mathrm{\pi}} {2}\, r^{4}$

 

[Femme_physics]

Erm...
$${T/r \over \pi r^2} = {300 Nm / 0.015 m \over \pi (0.015 m)^2} = 28.3 MPa \ne 56.6 MPa$$

Is this the formula you need to use to solve it? Because I don't see it in my formula sheet.

 

[I like Serena]

Is this the formula you need to use to solve it? Because I don't see it in my formula sheet.

No, this is not the formula you need to solve it.

It was derived from your formula $\tau_s={F_{max} \over A_s}$ and I showed it was wrong.
You cannot use that formula here, since it applies to linear shear (by a regular force) and not to torsional shear (by rotational torque).The formula you should use is the one given by NascentOxygen in his last post.
This is effectively the same as the formula you showed in your answer to PhantomJay, except that you did not specify what $I_0$ was.

 

[Femme_physics]

This is effectively the same as the formula you showed in your answer to PhantomJay, except that you did not specify what $I_0$ was.

Moment of inertia of a non-hollow cylinder.

http://img444.imageshack.us/img444/9633/mm2vh.jpg


But I don't have the mass in the question, so am not sure how am I supposed to use it.

No, this is not the formula you need to solve it.

It was derived from your formula $\tau_s={F_{max} \over A_s}$ and I showed it was wrong.
You cannot use that formula here, since it applies to linear shear (by a regular force) and not to torsional shear (by rotational torque).


The formula you should use is the one given by NascentOxygen in his last post.

Still quite off

http://img404.imageshack.us/img404/2576/69366748.jpg

 

[I like Serena]

Moment of inertia of a non-hollow cylinder.

But I don't have the mass in the question, so am not sure how am I supposed to use it.

Are you sure it is the moment of inertia?
Isn't it the "polar" moment of inertia?

Still quite off

http://img404.imageshack.us/img404/2576/69366748.jpg

Suppose you moved the decimal point 6 places to the left?

 

[Femme_physics]

Are you sure it is the moment of inertia?
Isn't it the "polar" moment of inertia?

Oh yes, it's polar. *smacks forehead*

Ow...been a while since I smacked my forehead.

http://img443.imageshack.us/img443/734/offufs.jpg

Still off...darn it

Suppose you moved the decimal point 6 places to the left?

I didn't move anything. That's the result.

 

[I like Serena]

Oh yes, it's polar. *smacks forehead*

Ow...been a while since I smacked my forehead.

((WAITT! Redoing exercise))
Damn it! Still wrong. This one is wayyyyyyy off.

I didn't move anything. That's the result.

In your result you have the unit [MPa], but it should be [Pa].
You still need to convert to [MPa], that is, move the decimal point 6 places over.

 

[Femme_physics]

In your result you have the unit [MPa], but it should be [Pa].
You still need to convert to [MPa], that is, move the decimal point 6 places over.

How do I make sure to always get the result in MPa? Do I use mm instead of m? BTW I edited my post above.

 

[I like Serena]

How do I make sure to always get the result in MPa? Do I use mm instead of m? BTW I edited my post above.

Yep. That should do the trick.
In your post above, you should use mm.

 

[Femme_physics]

Alright, did it! Kinda rushed the diagrams though...but just for the record

http://img851.imageshack.us/img851/1930/lonigmar.jpg

 

[I like Serena]

Alright, did it! Kinda rushed the diagrams though...but just for the record

Yep. You've got it down!

http://paniagua.es/wp-content/uploads/2011/03/smileygafas-115x115.png

 

[Femme_physics]

a w00t w00t! Femme-Fizzics in the hizzle yo!

Thanks ILS :)

 

[I like Serena]

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