Resultant of two forces with angular separation – perpendicularity condition Two forces act at an angle of 120°. The larger force is 40 N. If the resultant is perpendicular to the smaller force, find the magnitude of the smaller force.

Introduction / Context:
Vector addition with angular separation often appears in statics and dynamics. A useful technique is to convert geometric constraints (like perpendicularity) into dot-product equations. Here, the resultant is perpendicular to one of the forces, which allows a straightforward algebraic solution.

Given Data / Assumptions:

• Forces P (smaller, unknown) and Q (larger = 40 N).
• Angle between P and Q is 120°.
• Resultant R = P + Q is perpendicular to P.
• All vectors are coplanar; use standard Euclidean geometry.

Concept / Approach:
Use the dot product: If R ⟂ P, then R · P = 0. Since R = P + Q, we have P · P + Q · P = 0. Evaluate Q · P via magnitudes and the included angle: Q P cos 120°.

Step-by-Step Solution:

Verification / Alternative check:
Law of cosines on triangle formed by vectors can also be used to find R and then check perpendicularity via Pythagoras on magnitudes; you will arrive at the same p = 20 N.

Why Other Options Are Wrong:

• 40 N or 80 N: do not satisfy the orthogonality condition when substituted.
• 10√3 N: common distractor from mixing 60°/120° relations; algebra disproves it.

Common Pitfalls:
Using 60° instead of 120° (supplementary angle confusion) or forgetting the negative cosine for obtuse angles.

Final Answer:
20 N