question_answer

The horizontal force and the force inclined at an angle \[{{60}^{o}}\] with the vertical, whose resultant is in vertical direction of       P kg, are                                 [IIT 1983]

A)             \[P,2P\]

B)             \[P,\,\,P\sqrt{3}\]

C)             \[2P,\,\,P\sqrt{3}\]

D)             None of these

Correct Answer: C

Solution :

           Let \[\overrightarrow{OA}={{P}_{1}}\mathbf{i},\] \[\overrightarrow{CB}=-{{P}_{1}}\mathbf{i},\] \[\overrightarrow{OB}=-{{P}_{1}}\mathbf{i}+P\mathbf{j}\] \[\frac{\overrightarrow{OB}\,.\,j}{OB}=\cos 60{}^\circ \Rightarrow \frac{(-{{P}_{1}}i+Pj)\,.\,j}{\sqrt{P_{1}^{2}+{{P}^{2}}}}=\frac{1}{2}\]                    \[\Rightarrow 2P=\sqrt{{{P}^{2}}+P_{1}^{2}}\Rightarrow {{P}_{1}}=P\sqrt{3}\]                                   \[|\overrightarrow{OB}|\,=\sqrt{{{P}^{2}}+P_{1}^{2}}=\sqrt{{{P}^{2}}+3{{P}^{2}}}=2P.\]