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  • A circular disc of radius '3cm' has a hole of radius '2cm' at its centre as shown in the figure. If the surface m

A circular disc of radius '3cm' has a hole of radius '2cm' at its centre as shown in the figure. If the surface mass density of the disc varies as (\frac{20}{r}), where a_{0} is a positive constant and 'r' is the distance from the centre. The radius of gyration of such system is given by 'k', then the value of k^{2} will be :-

 

Option: 1

3 \mathrm{~cm}^{2}


Option: 2

6 \mathrm{~cm}^{2}


Option: 3

9 \mathrm{~cm}^{2}


Option: 4

18 \mathrm{~cm}^{2}


Answers (1)

best_answer

moment of inertia of the system about the central axis will be t

I=\left(\frac{a_{0}}{3}\right)(2 \pi)\left(b^{3}-a^{3}\right) \\

I=M K^{2} \\

K^{2}=\frac{I}{M}=\frac{\left(\frac{a_{0}}{3}\right)(2 \pi)\left(b^{3}-a^{3}\right)}{\left(a_{0}\right)(2 \pi)(b-a)}

k^{2}=\frac{\left(a^{2}+b^{2}+a b\right)}{3} \\

k^{2}=\frac{(3)^{2}+(2)^{2}+(2)(3)}{3} \\

k^{2}=\frac{9+4+6}{3}=6 \\

k^{2}=6

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