Question about the inertia matrix of a bar

In summary: So -β is the right answer.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.
  • #1
influx
164
2

Homework Statement


756c9e.png


Homework Equations


ml2/12

The Attempt at a Solution


5bf3a7.png
So according to my databook:

3267d3.png


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?
 
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  • #2
The moment of inertia for a bar around any axis perpendicular to it is ##m\ell^2/12##.
 
  • #3
Orodruin said:
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
 
  • #4
influx said:
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.
 
  • #5
Orodruin said:
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?
 
  • #6
influx said:
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.
 
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  • #7
Orodruin said:
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?
 
  • #8
I
influx said:
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?
 
  • #9
influx said:
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.
 

FAQ: Question about the inertia matrix of a bar

1. What is the inertia matrix of a bar?

The inertia matrix of a bar is a mathematical representation of its resistance to changes in rotational motion. It contains information about the bar's mass distribution and shape.

2. How is the inertia matrix of a bar calculated?

The inertia matrix is calculated by multiplying the mass matrix (a diagonal matrix with the bar's mass along the diagonal) by the square of the transformation matrix, which describes the orientation of the bar in space.

3. What does the inertia matrix tell us about a bar?

The inertia matrix provides information about how difficult it is to change the rotational motion of the bar. It includes the moments of inertia, which indicate how the mass is distributed around the rotation axis, and the products of inertia, which show how the mass is distributed off the rotation axis.

4. How does the inertia matrix affect the movement of a bar?

The inertia matrix determines the bar's resistance to changes in rotational motion. A higher inertia matrix means that it will be more difficult to change the bar's rotation, while a lower inertia matrix means it will be easier to change its rotation.

5. Can the inertia matrix of a bar change?

Yes, the inertia matrix of a bar can change if its mass distribution or shape is altered. For example, if weights are added to one end of the bar, the inertia matrix will increase, making it more difficult to change the bar's rotation.

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