How to find the maximum moment of a force about an axis?

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Discussion Overview

The discussion revolves around determining how to find the maximum moment of a force about a specified axis. Participants explore the relationship between the force vector components and the moment calculation, focusing on the implications of the force's direction and the geometry involved.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant suggests that to maximize the moment about the y-axis, the y-component of the force should be minimized, although they express uncertainty about this approach.
  • Another participant points out that the diagram indicates the force does not have a y-component, only x and z components, and questions the relationship between the vectors and the angles involved.
  • A participant calculates the components of the force vector based on given angles, providing specific values for Fx, Fy, and Fz, and presents the moment calculation about the axis.
  • One participant acknowledges a misunderstanding regarding the force's components after reviewing the diagram and emphasizes the need to focus on maximizing the moment without being constrained by the initial angles.
  • Another participant proposes manipulating the force vector to align it in the x-z plane for further calculations, suggesting a method to derive the angle α for maximizing the moment.
  • A later reply indicates that the proposed method may be overly complicated and redirects attention to a previous question about the magnitude of the cross product.

Areas of Agreement / Disagreement

Participants express differing views on how to approach the problem of maximizing the moment, with no consensus reached on the best method or the implications of the force vector's components.

Contextual Notes

Some participants reference specific angles and components of the force vector, but there is uncertainty regarding the correct interpretation of the diagram and how it affects the calculations. The discussion includes various assumptions about the relationships between the vectors and their components.

marsupialofdeath
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Homework Statement


Calculating the moment of force F about the axis isn't hard; how do I determine how to produce the maximum moment?
StaticsHW.png


Homework Equations


Maxis = ûaxis ⋅ (r×F)
F = 30N (This doesn't change at all during the problem)

The Attempt at a Solution


I would guess that since the problem axis is parallel to the y-axis, you should minimize that component of the force (β=90°). I'm not totally sure on that point though. As for the other two axes, I have no idea.
 
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marsupialofdeath said:
I would guess that since the problem axis is parallel to the y-axis, you should minimize that component of the force
That is certainly true, but the diagram does not even suggest F having such a component. It only shows x and z components.
What is the magnitude of rxF in terms of the magnitudes of the vectors and the angle between them?
 
haruspex said:
That is certainly true, but the diagram does not even suggest F having such a component. It only shows x and z components.
What is the magnitude of rxF in terms of the magnitudes of the vectors and the angle between them?
It's kind of hard to see in the diagram, but F is 60° from both the x and y axes.
Fx = F*cos(α) = 30*cos(60°) = 15 N
Fy = F*cos(β) = 30*cos(60°) = 15 N
Fz = F*cos(γ) = 30*cos(45°) ≈ 21.21 N
So, the vector for F should be (15i + 15j + 21.21k) N

And the vector for r is (-0.1i + 0.15k) m
So, r×F is (2.25i + 4.37j - 1.5k) N⋅m... That is the moment of force about the center of the given axis-- dot that with the unit vector j to determine that with the current angles, the moment about the axis is just 4.37 N⋅m. And we're trying to maximize that, so we need to maximize the j component of the r×F vector.

I'm not completely sure what you are asking, but I hope that clears things up a bit.
 
First, I withdraw my previous remark. On enlarging the diagram I see that it shows F as having a y component.
However, the question asks for the directions that would maximise the moment for the given magnitude. That surely means you should ignore the given angles for this part of the question.
 
haruspex said:
That surely means you should ignore the given angles for this part of the question.
Yeah, the first part of the question asked for the moment about the axis, which is what those angles were used for. So we need to manipulate the force vector so that instead of the given angles, it's in the x-z plane. So can we set it up so that Fx = 30cos(α) and Fz = 30cos(γ) = 30sin(α)? From there maybe we can do r×F again and derive it and set that equal to zero so we can solve for α.
 
marsupialofdeath said:
From there maybe we can do r×F again and derive it and set that equal to zero so we can solve for α.
That would work, but it is unnecessarily complicated.
Try to answer the question I asked at the end of post #2.
 

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