question_answer

An L-shaped object, made of thin rods of uniform mass density, is suspended with a string as shown in figure. If \[\,AB=BC\], and the angle made by AB with downward vertical is \[\theta \], then: [JEE Main Online Paper (Held On 09-Jan-2019 Morning]

A) \[\tan \,\theta \,\,=\,\,\frac{1}{3}\]       

B) \[\tan \,\theta \,\,=\,\,\frac{1}{2}\]

C) \[\tan \,\theta \,\,=\,\,\frac{2}{\sqrt{3}}\]     

D) \[\tan \,\theta \,\,=\,\,\frac{1}{2\sqrt{3}}\]

Correct Answer: A

Solution :

\[mg\text{ }\frac{a}{2}\,{{d}_{1}}=\,\,mg{{d}_{2}}\] \[mg\frac{a}{2}\,\sin \,\theta \,\,\,=\,\,\,mg\,(\frac{a}{2}\,\cos \,\theta \,\,-\,\,a\sin \,\theta )\] \[\frac{\sin \,\theta }{2}\,\,=\,\,\frac{\cos \,\theta }{2}\,\,-\,\sin \,\theta \] \[\frac{3}{2}\,\sin \,\theta \,\,=\,\,\frac{\cos \,\theta }{2}\] \[\tan \,\theta \,\,=\,\,\frac{1}{3}\]