Math Problem Statement
Choose correct
Solution
To solve this problem, let's go through each step carefully.
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Identify the Given Vector and Points:
- The force vector is given by .
- Point (the point where the force acts) is .
- Point (about which the moment is calculated) is .
- The moment vector about point is given as .
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Moment Calculation Formula: The moment of a force about a point is given by: where is the position vector from point to point , and is the force vector.
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Find :
- The position vector from to is:
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Calculate the Moment : The cross product in two dimensions (resulting in a moment along ) can be computed as: where and .
Substitute:
Then,
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Equate to the Given Moment: Since we know , we can set up the equation:
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Solve for :
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Conclusion: The correct answer is: Therefore, the answer is (c) .
Would you like more details on any of the steps, or have any questions?
Here are some related questions that expand on this concept:
- How is the direction of the cross product determined in three dimensions?
- What is the physical significance of a moment vector in mechanics?
- How would the calculation change if acted at a different point than ?
- What if the moment vector was given in terms of other directions (e.g., or )?
- How can we visualize a cross product in two dimensions?
Tip: Always keep track of the direction of vectors, especially when computing moments, as incorrect directions can lead to sign errors.
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Math Problem Analysis
Mathematical Concepts
Vectors
Moments
Cross Product
Formulas
Moment of a force about a point: M = r × F
Cross product in two dimensions: M = (x₁y₂ - y₁x₂)k̂
Theorems
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Suitable Grade Level
Undergraduate - Engineering or Physics
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