Question

two forces of 10N are acting at a point ,find the angle between them when the resultant magnitude is also equal to the magnitude of either of these forces
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Answer 1

        $\text{Let the angle between the two forces be }\theta$

        $\bar{F}_1 = 10 \:\hat{i}$

        $\bar{F}_{2} = 10cos\theta \: \hat{i} + 10sin\theta \: \hat{j}$

$\implies \: |\bar{F}_1 + \bar{F_2}| = 10$

$\implies \: \sqrt{(10cos\theta + 10)^2 + (10sin\theta)^2} = 10$

$\implies \: (10cos\theta + 10)^2 + (10sin\theta)^2 = 100$$\implies \: 100cos^2\theta + 100 + 200cos\theta + 100sin^2\theta = 100$

$\implies \: 100(sin^2\theta+ cos^2\theta) + 200cos\theta = 100 - 100$

$\implies \: 100(1) + 200cos\theta = 0$

$\implies \: 200cos\theta = -100$

$\implies \: cos\theta = -\dfrac{1}{2}$

$\implies \: \boxed{\theta = \dfrac{2\pi}{3}, \dfrac{4\pi}{3}}$