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KVPY Sample Paper KVPY Stream-SX Model Paper-12

  • question_answer
    A thin circular plate of mass M and radius R has its density varying as \[P(r)={{P}_{0}}r\] with \[{{P}_{0}}\] as Constant and \[r\] is the distance from its center. The moment of inertia of the circular plate about an axis perpendicular to the plate and passing through its edge is \[I=aM{{R}^{2}}.\]The value of the coefficient a is:

    A) \[\frac{3}{2}\]

    B) \[\frac{1}{2}\]

    C) \[\frac{8}{5}\]

    D) \[\frac{3}{5}\]

    Correct Answer: C

    Solution :

    \[M=\int_{0}^{R}{(\rho _{0}^{r}(2\pi rdr)=2\pi {{\rho }_{0}}\frac{{{R}^{3}}}{3}}\] \[I=M{{R}^{2}}+\int\limits_{0}^{R}{2\pi {{\rho }_{0}}{{r}^{4}}dr}=M{{R}^{2}}+\frac{2\pi {{\rho }_{0}}{{R}^{5}}}{5}\] \[I=M{{R}^{2}}+\frac{3M{{R}^{2}}}{5}=\frac{8M{{R}^{2}}}{5}.\]


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