Resultant of forces meeting at a point — compute magnitude from components If several forces act at a point, the magnitude of the single resultant R in terms of the algebraic sums of horizontal and vertical components (ΣH and ΣV) is:
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AR = √( (ΣH)^2 + (ΣV)^2 )
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BR = (ΣH)^2 + (ΣV)^2
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CR = (ΣH + ΣV)^2
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DR = ΣH + ΣV
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ER = √(ΣH + ΣV)
Answer
Correct Answer: R = √( (ΣH)^2 + (ΣV)^2 )
Explanation
Introduction / Context:Vector addition of forces is fundamental. When multiple forces act at a point, we commonly resolve them into orthogonal components and then recombine to find a single equivalent resultant force.
Given Data / Assumptions:
- All forces are concurrent (meet at a single point).
- Components are resolved along mutually perpendicular axes (horizontal H and vertical V).
- ΣH and ΣV denote algebraic sums of components.
Concept / Approach:The resultant of orthogonal components follows the Pythagorean relationship. If R has components (ΣH, ΣV), then the magnitude of R is given by R = sqrt( (ΣH)^2 + (ΣV)^2 ). The direction θ satisfies tan(θ) = ΣV / ΣH.
Step-by-Step Solution:
Resolve each force F_i into H and V components.Add components: ΣH = ΣF_ix and ΣV = ΣF_iy.Compute magnitude: R = √( (ΣH)^2 + (ΣV)^2 ).Optionally find direction: θ = atan2(ΣV, ΣH).Verification / Alternative check:Check special case: If ΣH = 0, then R = |ΣV|; if ΣV = 0, then R = |ΣH|. This is consistent with the square root expression.
Why Other Options Are Wrong:
- (ΣH)^2 + (ΣV)^2 without a square root: Gives the squared magnitude, not the magnitude.
- (ΣH + ΣV)^2 or ΣH + ΣV: Mixing components algebraically destroys vector nature.
- √(ΣH + ΣV): Nondimensional and incorrect.
Common Pitfalls:Forgetting to take the square root; sign errors when summing components; incorrect angle quadrant due to signs of ΣH and ΣV.
Final Answer:R = √( (ΣH)^2 + (ΣV)^2 )