A) \[\omega \propto \frac{1}{{{n}^{3}}}\]
B) \[\omega \propto \frac{1}{{{n}^{2}}}\]
C) \[\omega \propto {{n}^{2}}\]
D) \[\omega \propto {{n}^{3}}\]
Correct Answer: A
Solution :
Velocity, \[v\propto \frac{1}{n}\] \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,v=\frac{k}{n}\] Also, \[\operatorname{v} =r\omega \] So, \[r\omega =\frac{k}{n}\,\,\,\,\,\,\,\,\Rightarrow \,\,\,\,\,\,\omega =\frac{k}{nr}\] Now, radius, \[\operatorname{r}\propto {{n}^{2}}\] \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,r=k'{{n}^{2}}\] So, \[\omega =\frac{k}{n(k'{{n}^{2}})}=\frac{k}{k({{n}^{3}})}\] \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\omega \propto \frac{1}{{{n}^{3}}}\]
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