• # question_answer If$kr+r\times a=b,$ where $k$ is non-zero scalar and a,b are two given vectors, then will be 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)  $\frac{1}{{{k}^{2}}-{{a}^{2}}}\,\left( kb-\frac{a\cdot b}{k}-a\times b \right)$

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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