A) -1
B) 2
C) zero
D) 1
Correct Answer: A
Solution :
Key Concept When the resultant external torque acting on a system is zero, the total angular momentum of a system remains constant. This is the principle of the conservation of angular momentum. Given, force \[\mathbf{F}=\alpha \mathbf{\hat{i}}+3\mathbf{\hat{j}}+6\mathbf{\hat{k}}\]is acting at a point \[r=2\mathbf{\hat{i}}-6\mathbf{\hat{j}}-12\mathbf{\hat{k}}\] As, angular momentum about origin is conserved. i.e. \[\tau =\] constant Torque, \[\tau =0\Rightarrow \mathbf{r}\times \mathbf{F}=0\] \[\left| \begin{matrix} {\mathbf{\hat{i}}} & {\mathbf{\hat{j}}} & {\mathbf{\hat{k}}} \\ 2 & -6 & -12 \\ \alpha & 3 & 6 \\ \end{matrix} \right|=0\] \[\Rightarrow \,\,(-36+36)\mathbf{\hat{i}}-(12+12\alpha )\mathbf{\hat{j}}+(6+6\alpha )\mathbf{\hat{k}}=0\] So value of a for angular momentum about origin is conserved, \[\alpha =-1\]
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