Statement I: Two forces \((\vec{P} + \vec{Q})\) and \((\vec{P} - \vec{Q})\) where \((\vec{P} ⊥ \vec{Q})\), when act at an angle θ1 to each other, the magnitude of their resultant is \(\sqrt{3(P^2 + Q^2)}\), when they act at an angle θ2, the magnitude of their resultant becomes \(\sqrt{2(P^2 + Q^2)}\). This is possible only when θ1 < θ2.
Statement II: In the situation given above. θ1 = 60° and θ2 = 90°
In the light of the above statements, choose the most appropriate answer from the options given below:
(1) Statement-I is false but Statement-II is true
(2) Both Statement-I and Statement-II are true
(3) Statement-I is true but Statement-II is false
(4) Both Statement-I and Statement-II are false.
Correct answer: (2) Both Statement-I and Statement-II are true
Explanation:
\(\vec{A} = \vec{P} + \vec{Q}\)
\(\vec{B} = \vec{P} - \vec{Q}\)
\(\vec{P} ⊥ \vec{Q}\)
\(|\vec{A}| = |\vec{B}|\) = \(\sqrt{P^2 + Q^2}\)
\(|\vec{A} + \vec{B}|\) = \(\sqrt{2(P^2 + Q^2)(1 + cosθ)}\)
For \(|\vec{A} + \vec{B}|\) = \(\sqrt{3(P^2 + Q^2)}\)
θ1 = 60°
For \(|\vec{A} + \vec{B}|\) = \(\sqrt{2(P^2 + Q^2)}\)
θ2 = 90°