A) \[\frac{1}{{{k}^{2}}+{{a}^{2}}}\,\left( kb+\frac{a\cdot b}{k}a+a\times b \right)\]
B) \[\frac{1}{{{k}^{2}}+{{a}^{2}}}\,\left( kb-\frac{a\cdot b}{k}a+a\times b \right)\]
C) \[\frac{1}{{{k}^{2}}+{{a}^{2}}}\,\left( kb-\frac{a\cdot b}{k}a-a\times b \right)\]
D) \[\frac{1}{{{k}^{2}}-{{a}^{2}}}\,\left( kb-\frac{a\cdot b}{k}-a\times b \right)\]
Correct Answer: A
Solution :
\[W=\mathbf{T}\cdot \mathbf{d}\Rightarrow \,W=Td\] ?(i) Taking dot product with a in Eq. (i) \[\Rightarrow \] \[W=-Td=-\frac{3Mgd}{4}\] \[\Sigma mvr=\,({{l}_{system}})\omega \] ?(ii) \[\Rightarrow \] Taking cross product with a in Eq. (i), we get \[mv\frac{l}{2}=\frac{(2m)\,{{(2l)}^{2}}}{12}\omega =\frac{2m(4{{l}^{2}})}{12}\omega \] \[\Rightarrow \] \[\omega =\frac{3v}{4l}\] [from Eqs. (i) and (ii)] \[T=2\pi \,\sqrt{\frac{L}{g}}\] \[\frac{L}{2}\] \[\Rightarrow \] \[T=2\pi \sqrt{\frac{L}{2g}}\]
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