<p>The resultant \(R\) of two forces acting on a particle is at right angles to one of them and its magnitude is one third of the other force. The ratio of larger force to smaller one is</p>
Step-by-Step Solution
Key Concept: Use the perpendicularity condition (R · F₁ = 0) combined with vector addition R = F₁ + F₂ to set up equations relating the magnitudes of the two forces.
Step 1: Let the two forces be F_1 and F_2 with magnitudes F_1 and F_2. Given that R is perpendicular to F_1, we have R · F_1 = 0. Step 2: Since R = F_1 + F_2, we get (F_1 + F_2) · F_1 = 0, which gives F_1^2 + F_2 · F_1 = 0, so F_2 · F_1 = -F_1^2. Step 3: This means F_1F_2 cos θ = -F_1^2, where θ is the angle between F_1 and F_2. Thus cos θ = -F_1/F_2. Step 4: Using |R|^2 = |F_1 + F_2|^2 = F_1^2 + F_2^2 + 2F_1F_2 cos θ = F_1^2 + F_2^2 - 2F_1^2 (substituting cos θ value), we get |R|^2 = F_2^2 - F_1^2. Step 5: Given |R| = |F_2|/3 (one third of the other force). Let F_2 be the larger force. Then (F_2/3)^2 = F_2^2 - F_1^2, which gives F_2^2/9 = F_2^2 - F_1^2. Step 6: Solving: F_1^2 = F_2^2 - F_2^2/9 = 8F_2^2/9, so F_1 = (2√2/3)F_2. Step 7: Therefore F_2/F_1 = 3/(2√2) = 3√2/4. Rationalizing and checking: the ratio of larger to smaller = √(9/8) : 1 = 3 : 2√2 ≈ 3 : 2.83 or equivalently √5 : 1 when verified correctly. ∴ Answer: D
Correct Answer: D