Questions: Force Vectors, Components, and Resultants

Question 1

A 200 N force acts at 30° above the horizontal. What is its horizontal component?

• A. 200 N (the full magnitude acts horizontally)
• B. 100 N (sin 30° × 200)
• C. 173 N (cos 30° × 200)
• D. 141 N (cos 45° × 200)

Answer: (C)

The horizontal component uses the cosine of the angle from horizontal: Fx = 200 cos 30° = 200 × 0.866 ≈ 173 N. The vertical component is Fy = 200 sin 30° = 200 × 0.5 = 100 N. A common error is swapping sine and cosine: cos gives the adjacent side (horizontal when angle is measured from horizontal), sin gives the opposite side (vertical). The two components are not the force itself — they are its projections onto the coordinate axes, and Fx² + Fy² = F².

Question 2

Three forces act on a ring: 10 N east, 6 N west, and 8 N north. What is the magnitude of the resultant force?

• A. 24 N (sum of all magnitudes)
• B. 8.9 N
• C. 4 N (net east-west component only)
• D. 14 N (10 N + 8 N - 6 N, treating all as positive)

Answer: (B)

Sum components separately: Rx = 10 - 6 = 4 N (east), Ry = 8 N (north). Resultant magnitude R = √(Rx² + Ry²) = √(16 + 64) = √80 ≈ 8.9 N. Adding magnitudes directly (24 N) is wrong because it ignores direction — forces pointing in opposite directions partially cancel. The component method correctly handles direction by treating westward forces as negative x-components before summing.

Question 3

To find the net effect of multiple forces acting on a body, you can add their magnitudes directly.

• T. True
• F. False

Answer: False

False — force is a vector quantity; magnitude alone ignores direction. Adding magnitudes only gives the correct resultant if all forces point in the same direction (the maximum possible case). A 10 N force east and a 10 N force west have a resultant magnitude of 0 N, not 20 N. The correct procedure is to decompose each force into components (x, y, z), sum each set of components algebraically (respecting sign/direction), then compute the resultant magnitude and direction from the summed components.

Question 4

A body is in translational equilibrium when the vector sum of all forces acting on it equals zero.

• T. True
• F. False

Answer: True

True — this is the definition of translational equilibrium. ΣF = 0 means the resultant of all forces is the zero vector, which in component form requires ΣFx = 0, ΣFy = 0, and ΣFz = 0 simultaneously. Each is a separate scalar equation. A non-zero resultant would produce acceleration (F = ma), so equilibrium requires the resultant to vanish. This is why component decomposition is so powerful: it converts a single vector condition into a system of scalar equations that can be solved algebraically.

Question 5

Why is decomposing forces into components essential for equilibrium analysis, rather than working with magnitudes and angles directly?

Model answer: Decomposing forces into components converts the vector equilibrium condition ΣF = 0 into separate scalar equations: ΣFx = 0, ΣFy = 0, ΣFz = 0. Scalar equations can be added and subtracted algebraically — you can sum any number of forces by independently summing their x-components, y-components, and z-components, then reconstruct the resultant. Working directly with magnitudes and angles requires trigonometric constructions for each force combination and quickly becomes unmanageable with more than two forces. Components make superposition systematic and the algebra routine.

The principle of superposition — that the resultant produces the same external effect as the original system — is what justifies replacing multiple forces with their component sums. Equilibrium then becomes three independent algebraic conditions, each solvable for one unknown. This is why every statics problem (beams, trusses, pulleys) begins with a free-body diagram and component decomposition.