The resultant of two forces, magnitude of each is equal to ‘P’ and the acting angle between them is $60^{\circ}$, is
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For two equal forces $P$ at angle $\theta$, the resultant can be simplified to $R = 2P \cos(\theta/2)$.
For $\theta = 60^{\circ}$: $R = 2P \cos 30^{\circ} = 2P \left(\frac{\sqrt{3}}{2}\right) = \sqrt{3}P$. This shortcut saves time during exams!
1. The Law of Cosines for Resultant: For two forces $F_1$ and $F_2$ acting at an angle $\theta$, the magnitude of the resultant $R$ is given by:
$$R = \sqrt{F_1^2 + F_2^2 + 2F_1F_2 \cos \theta}$$
2. Applying Given Values: We are given that:
• $F_1 = P$
• $F_2 = P$
• $\theta = 60^{\circ}$
Substituting these into the formula:
$$R = \sqrt{P^2 + P^2 + 2(P)(P) \cos 60^{\circ}}$$
3. Mathematical Calculation: Since $\cos 60^{\circ} = \frac{1}{2}$:
$$R = \sqrt{2P^2 + 2P^2 \left(\frac{1}{2}\right)}$$
$$R = \sqrt{2P^2 + P^2}$$
$$R = \sqrt{3P^2}$$
$$R = \sqrt{3} \text{ P}$$
The magnitude of the resultant force is exactly $\sqrt{3}$ times the magnitude of the individual forces.