Three concurrent forces produce ΣFx = +4 kN and ΣFy = −3 kN after component decomposition. What is the resultant's magnitude and which quadrant does it point into?
A3.5 kN, pointing into the first quadrant (up and right)
B7 kN, pointing into the fourth quadrant (right and down)
C5 kN, pointing into the fourth quadrant (right and down)
D1 kN, pointing into the second quadrant (left and up)
The resultant magnitude is |R| = √(ΣFx² + ΣFy²) = √(16 + 9) = √25 = 5 kN. The positive ΣFx (rightward) and negative ΣFy (downward) place the resultant in the fourth quadrant. Option A incorrectly adds the component magnitudes directly (4 − 3 = 1 or averages). Option B adds them directly (4 + 3 = 7). Only the Pythagorean theorem applied to components gives the correct magnitude.
Question 2 Multiple Choice
A force of 200 N acts at 30° above the horizontal. A student computes Fx = 200 sin 30° = 100 N and Fy = 200 cos 30° = 173 N. What error did the student make?
ANo error — sine and cosine can be used interchangeably with angles measured from the horizontal
BThe magnitude 200 N should be doubled before applying trigonometry
CThe trig functions are swapped — Fx uses cosine (adjacent/hypotenuse) and Fy uses sine when the angle is measured from the x-axis
DThe force must be decomposed in 3D even for 2D problems
When angle θ is measured from the positive x-axis (the horizontal), the x-component is the adjacent side: Fx = F cos θ, and the y-component is the opposite side: Fy = F sin θ. The student reversed the functions. For θ = 30°: Fx = 200 cos 30° ≈ 173 N (rightward) and Fy = 200 sin 30° = 100 N (upward). The student's values produce a force that is mostly vertical rather than mostly horizontal — which is incorrect for a nearly-horizontal force at 30°. This confusion between which trig function applies to which component is the most common decomposition error.
Question 3 True / False
The resultant of a system of concurrent forces is mechanically equivalent to the original forces — a body cannot distinguish between experiencing all the individual forces and experiencing only their resultant.
TTrue
FFalse
Answer: True
This equivalence is the foundational principle behind the method. A rigid body subject to five concurrent forces has the same translational behavior as one subject to a single resultant force of equal magnitude and direction. This is why collapsing a force system to its resultant is valid for equilibrium analysis: when you write ΣFx = 0 and ΣFy = 0, you are demanding the resultant equal zero — which is equivalent to demanding no net translational effect from all the individual forces combined.
Question 4 True / False
To find the resultant magnitude of two concurrent forces, you can simply add their magnitudes together.
TTrue
FFalse
Answer: False
Magnitudes cannot be added directly unless the forces are parallel and in the same direction. Forces are vectors; their resultant is found by vector addition — decomposing into components, summing each component algebraically, then recombining with the Pythagorean theorem. For two forces of 3 N and 4 N at right angles, the resultant is 5 N, not 7 N. Adding magnitudes (3 + 4 = 7) gives the maximum possible resultant (when forces are parallel and same direction) and is generally incorrect. The actual resultant depends on both magnitudes and the angle between the forces.
Question 5 Short Answer
Why does the method of component decomposition work, and why is it preferred over graphical vector addition when three or more forces are involved?
Think about your answer, then reveal below.
Model answer: Component decomposition works because any force vector can be resolved into its projections onto orthogonal axes using trigonometry — specifically, Fx = F cos θ and Fy = F sin θ for a 2D force at angle θ from the x-axis. Once decomposed, x-components and y-components are independent scalars that can be summed algebraically. The resultant is then recovered using the Pythagorean theorem and arctangent. This is preferred over graphical addition for three or more forces because graphical methods (tip-to-tail drawing) accumulate geometric errors with each additional force and become unwieldy with arbitrary angles. The algebraic method is exact and scales to any number of forces.
The deeper reason it works is that vector addition is commutative and associative — you can add vectors in any order, and decomposition preserves this. Each axis is treated independently, which reduces a 2D vector problem to two 1D scalar problems. This is a general strategy in physics and engineering: resolve a multi-dimensional problem into independent components, solve each, then recombine.