Question

Statement I : Two forces \((\vec{p} + \vec{q})\) and \((\vec{p} - \vec{q})\) where \(\vec{p} \perp \vec{q}\), when act at an angle \(\theta_1\) to each other, the magnitude of their resultant is \(\sqrt{3(p^2 + q^2)}\), when they act at an angle \(\theta_2\), the magnitude of their resultant becomes \(\sqrt{2(p^2 + q^2)}\). This is possible only when \(\theta_1<\theta_2\).
Statement II : In the situation given above, \(\theta_1 = 60^{\circ}\) and \(\theta_2 = 90^{\circ}\).
In the light of the above statements, choose the most appropriate answer from the options given below :

• A. Both Statement I and Statement II are true.
• B. Both Statement I and Statement II are false.
• C. Statement I is true but Statement II is false.
• D. Statement I is false but Statement II is true.

Hint:

If two vectors have equal magnitude \(A\), their resultant is \(2A \cos(\theta/2)\). In this problem, both vectors have magnitude \(\sqrt{p^2+q^2}\) because they are perpendicular diagonals of a rectangle formed by \(p\) and \(q\).

Answer 1

The Correct Option is A

To resolve the question, we need to analyze the statements based on vector addition and resultant force. Given the forces \((\vec{p} + \vec{q})\) and \((\vec{p} - \vec{q})\), and knowing that these vectors are orthogonal (\(\vec{p} \perp \vec{q}\)), we will use the formula for the magnitude of the resultant of two forces:

\(R = \sqrt{A^2 + B^2 + 2AB\cos\theta}\)

where \(A = \|\vec{p} + \vec{q}\|\) and \(B = \|\vec{p} - \vec{q}\|\).

1. For Statement I: When the resultant is \(\sqrt{3(p^2 + q^2)}\), we have:
   • \(R_1 = \sqrt{3(p^2 + q^2)}\)
   • \(R_1 = \sqrt{2(p^2 + q^2) + 2pq\cos\theta_1}\)
   • \(3(p^2 + q^2) = 2(p^2 + q^2) + 2pq\cos\theta_1\)
   • \(pq\cos\theta_1 = (p^2 + q^2)/2\)
2. For Resultant \(\sqrt{2(p^2 + q^2)}\), using a similar process:
   • \(R_2 = \sqrt{2(p^2 + q^2)}\)
   • \(R_2 = \sqrt{2(p^2 + q^2) + 2pq\cos\theta_2}\)
   • \(2(p^2 + q^2) = 2(p^2 + q^2) + 2pq\cos\theta_2\)
   • \(pq\cos\theta_2 = 0\)
   • This implies \(\theta_2 = 90^\circ\).
3. Hence, based on the given conditions:
   • From \(\theta_1\): \(<\theta_2\), and the geometry of vectors:
     • For \(\theta_1\) solution, \(\theta_1 = 60^\circ\).

Thus, both statements are consistent:

1. Statement I asserts the condition of angles during the resultant calculations are consistent.
2. Statement II correctly assigns \(\theta_1 = 60^\circ\) and \(\theta_2 = 90^\circ\).

Conclusion: Both Statement I and Statement II are true. Therefore, the correct answer is:

Both Statement I and Statement II are true.