Math Problem Statement

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If F = L î - 2 ĵ acts at the point A(5, 2) and the moment vector of F about the point B(7, -4) equals 20 k̂, then the value of L is...

Solution

To solve this problem, let's go through each step carefully.

  1. Identify the Given Vector and Points:

    • The force vector is given by F=Li^2j^\vec{F} = L \hat{i} - 2 \hat{j}.
    • Point AA (the point where the force acts) is (5,2)(5, 2).
    • Point BB (about which the moment is calculated) is (7,4)(7, -4).
    • The moment vector MB\vec{M}_B about point BB is given as 20k^20 \hat{k}.
  2. Moment Calculation Formula: The moment of a force about a point BB is given by: MB=rAB×F\vec{M}_B = \vec{r}_{AB} \times \vec{F} where rAB\vec{r}_{AB} is the position vector from point BB to point AA, and F\vec{F} is the force vector.

  3. Find rAB\vec{r}_{AB}:

    • The position vector rAB\vec{r}_{AB} from B(7,4)B(7, -4) to A(5,2)A(5, 2) is: rAB=(57)i^+(2(4))j^=2i^+6j^\vec{r}_{AB} = (5 - 7) \hat{i} + (2 - (-4)) \hat{j} = -2 \hat{i} + 6 \hat{j}
  4. Calculate the Moment MB=rAB×F\vec{M}_B = \vec{r}_{AB} \times \vec{F}: The cross product in two dimensions (resulting in a moment along k^\hat{k}) can be computed as: MB=(x1y2y1x2)k^\vec{M}_B = (x_1 y_2 - y_1 x_2) \hat{k} where rAB=x1i^+y1j^\vec{r}_{AB} = x_1 \hat{i} + y_1 \hat{j} and F=x2i^+y2j^\vec{F} = x_2 \hat{i} + y_2 \hat{j}.

    Substitute:

    • rAB=2i^+6j^\vec{r}_{AB} = -2 \hat{i} + 6 \hat{j}
    • F=Li^2j^\vec{F} = L \hat{i} - 2 \hat{j}

    Then, MB=(2)(2)(6)(L)=46L\vec{M}_B = (-2)(-2) - (6)(L) = 4 - 6L

  5. Equate to the Given Moment: Since we know MB=20k^\vec{M}_B = 20 \hat{k}, we can set up the equation: 46L=204 - 6L = 20

  6. Solve for LL: 6L=204-6L = 20 - 4 6L=16-6L = 16 L=166=83L = -\frac{16}{6} = -\frac{8}{3}

  7. Conclusion: The correct answer is: 83\boxed{-\frac{8}{3}} Therefore, the answer is (c) 83-\frac{8}{3}.

Would you like more details on any of the steps, or have any questions?

Here are some related questions that expand on this concept:

  1. How is the direction of the cross product determined in three dimensions?
  2. What is the physical significance of a moment vector in mechanics?
  3. How would the calculation change if F\vec{F} acted at a different point than AA?
  4. What if the moment vector was given in terms of other directions (e.g., i^\hat{i} or j^\hat{j})?
  5. 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