Two forces and are acting on a body. One force has magnitude thrice that of the other force and the resultant of the two forces is equal to the force of larger magnitude. The angle between and is The value of is

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Solution

Q.
Two forces ${\vec{F}}_{1}$ and ${\vec{F}}_{2}$ are acting on a body. One force has magnitude thrice that of the other force and the resultant of the two forces is equal to the force of larger magnitude. The angle between ${\vec{F}}_{1}$ and ${\vec{F}}_{2}$ is $\cos^{- 1} (\frac{1}{n}) .$ The value of $| n |$ is ___________ . [2024]

Ans.
(6) $let F_{1} = F_{0} then, F_{2} = 3 F_{0} also, F_{R} = 3 F_{0}$

$F_{R}^{2} = F_{1}^{2} + F_{2}^{2} + 2 F_{1} F_{2} \cos θ$

$∴ 3 F_{0} = \sqrt{F_{0}^{2} + 9 F_{0}^{2} + 6 F_{0}^{2} \cos θ}$

$3 F_{0} = F_{0} \sqrt{10 + 6 \cos θ}$

$9 = 10 + 6 \cos θ$

$\cos θ = - \frac{1}{6} \Rightarrow θ = \cos^{- 1} (\frac{1}{- 6})$

$n = - 6 \Rightarrow | n | = 6$

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