<p>From the parallelogram law of forces, if \(R^2 = P^2 + Q^2 + 2PQ\cos\alpha\) and equations are set up for doubled forces, the ratio \(P^2 : Q^2 : R^2\) equals:</p>
Step-by-Step Solution
Key Concept: When forces are doubled in the parallelogram law formula, the resultant becomes 2R. By substituting 2P, 2Q, and 2R into the vector equation and comparing magnitudes, we find that the ratio of squared magnitudes remains invariant under uniform scaling.
Step 1: Start with the parallelogram law: R^2 = P^2 + Q^2 + 2PQ cos α Step 2: When forces are doubled, substitute 2P for P, 2Q for Q, and the resultant becomes 2R: (2R)^2 = (2P)^2 + (2Q)^2 + 2(2P)(2Q) cos α Step 3: Simplify: 4R^2 = 4P^2 + 4Q^2 + 8PQ cos α 4R^2 = 4(P^2 + Q^2 + 2PQ cos α) Step 4: Divide both sides by 4: R^2 = P^2 + Q^2 + 2PQ cos α This is identical to the original equation, proving the fundamental relationship is preserved. Step 5: Therefore, the ratio P^2 : Q^2 : R^2 remains unchanged when forces are uniformly scaled. ∴ Answer: B
Correct Answer: B