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The resultant of the forces \(P\) and \(Q\) is \(R\). If \(Q\) is doubled, the resultant also doubles in magnitude. Find the angle between \(P\) and \(Q\).

1. \(\cos\theta = \frac{Q}{2P}\)
2. \(\cos\theta = \frac{4Q-3P}{\,}\)
3. \(\cos\theta = \frac{2Q-3P}{\,}\)
4. \(\cos\theta = \frac{3P-4Q}{\,}\)

Correct answer: \(\cos\theta = \frac{3P-4Q}{\,}\)

Solution

Let the original resultant be \(R\), so \(R^2=P^2+Q^2+2PQ\cos\theta\). After doubling \(Q\), the new resultant has magnitude \(2R\), giving \(4R^2=P^2+4Q^2+4PQ\cos\theta\). Solving these equations yields the required relation for \(\cos\theta\).

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