Resultant of two coplanar forces: If two forces P and Q act at an angle θ, which expression corresponds to the square of the resultant magnitude?

Introduction / Context:

Combining two forces acting at an angle is a staple in engineering mechanics. The magnitude of the resultant can be obtained via the law of cosines applied to the force triangle or via vector components. Some MCQs list the expression for R^2 rather than R, so it is important to recognize both forms.

Given Data / Assumptions:

• Two forces P and Q act at an included angle θ (0° ≤ θ ≤ 180°).
• Planar (coplanar) force system.

Concept / Approach:

Construct the triangle of forces. By the law of cosines on the magnitude triangle, the resultant magnitude R satisfies R^2 = P^2 + Q^2 + 2PQ cos θ. Equivalently, from components, R_x = P + Q cos θ and R_y = Q sin θ if P is taken as the reference direction, and then R^2 = R_x^2 + R_y^2 simplifies to the same expression.

Step-by-Step Solution:

Verification / Alternative check:

Component method corroborates the same result: R^2 = (P + Q cos θ)^2 + (Q sin θ)^2 expands to P^2 + Q^2 + 2PQ cos θ.

Why Other Options Are Wrong:

• Terms with sin θ or tan θ do not arise from the magnitude relation.
• P + Q + 2PQ cos θ has wrong dimensions (mixes linear and quadratic terms).
• P^2 + Q^2 − 2PQ cos θ corresponds to the case where angle between P and (−Q) is θ, not the standard included angle here.

Common Pitfalls:

• Forgetting that many textbooks present R = sqrt(P^2 + Q^2 + 2PQ cos θ); this option lists the squared form.

Final Answer:

P^2 + Q^2 + 2PQ cos θ