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.
-
A20 N
-
B40 N
-
C80 N
-
D10√3 N
Answer
Correct Answer: 20 N
Explanation
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:
Let |P| = p and |Q| = 40.Condition: (P + Q) · P = 0 ⇒ P · P + Q · P = 0.Compute: P · P = p^2; Q · P = |Q||P| cos 120° = 40 p (−1/2) = −20 p.Equation: p^2 − 20 p = 0 ⇒ p(p − 20) = 0 ⇒ p = 20 N (reject p = 0).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