Find the magnitude of two forces such that when they act at right angles, their resultant is 5 N, and when they act at an angle of 60°, their resultant is √37 N.

Force Magnitude Problem Solution

Problem Statement

Given Conditions

Resultant at 90° (R₁) 5 N

Resultant at 60° (R₂) √37 N

Solution

Case 1: Orthogonal Forces (90°)

Step 1: Apply Pythagorean theorem:

R₁² = P² + Q²
5² = P² + Q²
25 = P² + Q² (Equation 1)

Case 2: 60° Angle Between Forces

Step 2: Apply cosine law:

R₂² = P² + Q² + 2PQ cos60°
(√37)² = P² + Q² + 2PQ(½)
37 = P² + Q² + PQ (Equation 2)

Solving the System

Step 3: Subtract equations:

37 – 25 = PQ ⇒ 12 = PQ

Step 4: Form quadratic equation:

Let P = 12/Q ⇒ (12/Q)² + Q² = 25
144/Q² + Q² = 25 ⇒ Q⁴ – 25Q² + 144 = 0

Step 5: Solve quadratic:

Let x = Q² ⇒ x² – 25x + 144 = 0
x = [25 ± √(625 – 576)]/2 = 16 or 9
⇒ Q = 4 N or 3 N

Final Solution: P = 3 N, Q = 4 N

Verification & Analysis

Orthogonal Verification: 3² + 4² = 9 + 16 = 25 ✔️
60° Verification: 3² + 4² + 3×4 = 9 + 16 + 12 = 37 ✔️

Physical Interpretation

The solution demonstrates fundamental vector addition principles. The 60° configuration’s resultant exceeds the orthogonal case due to constructive interference of force components, following the parallelogram law of vectors.