Two forces F_1 and 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 F_…

Question

Two forces F_1 and 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 F_1 and F_2 is cos^-1 ⁡ ( 1n ). The value of |n | is ________.

Accepted Answer

Correct answer: 6. (6) Let's denote the magnitude of the smaller force as F, hence the magnitude of the larger force is 3F. The resultant force R is equal in magnitude to the larger force, which means |R |=3F. When two forces F_1 and F_2 act on a body, the magnitude of their resultant R can be found using the law of vector addition: |R |= (F_1 )^2+ (F_2 )^2+2 (F_1 ) (F_2 ) cos ⁡ where is the angle between F_1 and F_2. Given that in our case |R |=3F , (F_1 )=F and (F_2 )=3F, by substituting these values into the equation, we get: 3F= F^2+ (3F )^2+2 (F ) (3F ) cos ⁡ 9F^2=F^2+9F^2+6F^2 cos ⁡ Simplifying this equation by subtracting 10F^2 from both sides gives: -F^2=6F^2 cos ⁡ Dividing both sides by -F^2 gives: -1=-6 cos ⁡ Therefore, cos ⁡ = 16. It is given that the angle between F_1 and F_2 is cos^-1 ⁡ ( 1n ), hence comparing this with the above result, we find that n=6. Therefore, |n |=6.

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

Solution

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