The resultant of two foces 3P and 2P is R. if the first is doubled the resultant is also doubled. Then , angle between two forces.

Explanation

By ` C^(2) =A^(2) +B^(2) = 2AB cos theta`
Case 1 : ` R^(2) = (3P)^(2) + (2P)^(2) + 2(3P) (2P) cos theta`
` Rightarrow R^(2) = 9P^(2) + 4P^(2) + 12 P^(2) cos theta`
` Rightarrow R^(2) = 13P^(2) + 12P^(2) cos theta`
Case 2: ` (2R)^(2) = (6P)^(2) + (2P)^(2) + 2(6P) (2P) cos theta `
` Rightarrow 4R^(2) = 36P^(2) + 4P^(2) + 24P^(2) cos theta`
`Rightarrow 4R^(2) = 40 P^(2) + 24 P^(2) cos theta`
Form equation (1) and (2)
` 4 {13 P^(2) + 12 P^(2) cos theta} = 40 P^(2) +24P^(2) cos theta`
` Rightarrow 53 P^(2)+ 48 P^(2) cos theta + 24 P^(2) cos theta`
` Rightarrow 24 P^(2) cos theta - 12 P^(2)`
` Rightarrow = - 12/24 = -1/2 = cos theta 120^(@) Rightarrow theta = 120^(@)`