Question about the inertia matrix of a bar

[influx]

Homework Statement

Homework Equations

ml²/12

The Attempt at a Solution

So according to my databook:

The axis x'-x' in the question corresponds to axis ZZ in the databook image above. That means in terms of radius, the moment of inertia about axis x'-x' is mr²/2. So in light of this, why is the constant outside the matrix ml²/12? Why not mr²/2?

 

[Orodruin]

The moment of inertia for a bar around any axis perpendicular to it is $m\ell^2/12$.

 

[influx]

The moment of inertia for a bar around any axis perpendicular to it is $m\ell^2/12$.

I understand that but the x'-x' axis in the question is parallel to the bar and not perpendicular?

Thanks

 

[Orodruin]

I understand that but the x'-x' axis in the question is parallel to the bar and not perpendicular?

Thanks

Exactly, and the moment of inertia relative to that axis is zero, as shown in the solution.

 

[influx]

Exactly, and the moment of inertia relative to that axis is zero, as shown in the solution.

If the moment of inertia relative to the parallel axis is zero, then why is the moment of inertia about axis ZZ (in my data book) not 0? This axis is parallel to the cylinder?

 

[Orodruin]

If the moment of inertia relative to the parallel axis is zero, then why is the moment of inertia about axis ZZ (in my data book) not 0? This axis is parallel to the cylinder?

Because the problem you are solving is idealising the rod as infinitely thin - i.e., having negligible diameter.

 

[influx]

Because the problem you are solving is idealising the rod as infinitely thin - i.e., having negligible diameter.

Ah that makes sense! This question is from an exam paper set by my university's engineering department a few years ago, is there any way to determine that this rod is infinitely thin from the wording of the question? Does the ''uniform slender'' part suggest this?

Also, I was wondering why (according to the mark scheme) the angle is -β? I mean the diagram shows a counter clock wise rotation so shouldn't the angle be β?

 

[influx]

I

Ah that makes sense! This question is from an exam paper set by my university's engineering department a few years ago, is there any way to determine that this rod is infinitely thin from the wording of the question? Does the ''uniform slender'' part suggest this?

Also, I was wondering why (according to the mark scheme) the angle is -β? I mean the diagram shows a counter clock wise rotation so shouldn't the angle be β?

I was wondering whether anyone could help me out with this?

 

[Orodruin]

Ah that makes sense! This question is from an exam paper set by my university's engineering department a few years ago, is there any way to determine that this rod is infinitely thin from the wording of the question? Does the ''uniform slender'' part suggest this?

Also, I was wondering why (according to the mark scheme) the angle is -β? I mean the diagram shows a counter clock wise rotation so shouldn't the angle be β?

So you need a negative angle to counter that offset. The y-axis is pointing into the screen so a positive rotation would typically be clockwise.