In the xy plane, a force [latex]F = F_{x}i + F_{y}j[/latex] of magnitude 10 acts along the line r = i + j + λ(3 i + 4 j ) in the direction of increasing λ . i) Find [latex]F_{x}[/latex] and [latex]F_{y}.[/latex] ii) F acts through the point A(a, 0). Find a. iii) Deduce the moment of F about O.

i) F is in the same direction as 3 i + 4 j . |3 i + 4 j | = √(3² + 4²) = 5 So the unit vector in the direction of F is [latex]\frac{1}{5}(3i + 4j).[/latex] Also |F| = 10, hence F = [latex]\frac{10}{5}(3i + 4j) = 6i + 8j[/latex] ⇒ [latex]F_{x} = 6 [/latex] and [latex]F_{y} = 8 [/latex] ii) r = i + j + λ(3 i + 4 j ) = (1 + 3λ) i + (1 + 4λ) j ⇒ x = 1 + 3λ and y = 1 + 4λ where ( a, y ) is a point on the line. When y = 0, λ = [latex]- \frac{1}{4}[/latex] ⇒ x = [latex] \frac{1}{4}[/latex]; hence a = [latex]\frac{1}{4}.[/latex] iii) To find the moment about O, regard the force as acting at A and resolved into its two components [latex]F_{x}[/latex] and [latex]F_{y}. F_{x}[/latex] does not contribute, so the result is a[latex]F_{y} = \frac{1}{4} × 8 = 2 [/latex] anticlockwise

Question 17.1: In the xy plane, a force F = Fxi + Fyj of magnitude 10 acts ...

An Introduction to Mathematics for Engineers: Mechanics

In the xy plane, a force $F = F_{x}i + F_{y}j$ of magnitude 10 acts along the line

r = i + j + λ(3i + 4j)

in the direction of increasing λ.
i) Find $F_{x}$ and $F_{y}.$
ii) F acts through the point A(a, 0). Find a.
iii) Deduce the moment of F about O.

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