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A circular disc of radius b has a hole o...

A circular disc of radius b has a hole of radius a at its centre (see figure). If the mass per unit area of the disc varies as `((sigma_(0))/(r))`. then the radius of gyration of the disc about its axis passing through the centre is:

A

`(a+b)/(2)`

B

`sqrt((a^(2)+b^(2)+ab)/(2))`

C

`sqrt((a^(2)+b^(2)+ab)/(3))`

D

`(a+b)/(3)`

Text Solution

Verified by Experts

The correct Answer is:
C
`I=intdm.r^(2)`
`I=int_(a)^(b)(sigma_(0))/(r)2pirdr.r^(2)`
`impliesI=sigma_(0)2pi((b^(3)-a^(3))/(3))`
`M=int_(a)^(b)dm=int_(a)^(b)(sigma_(0))/(r)2pi r dr=sigma_(0)2pi(b-a)`
`I=MK^(2)`
`sigma_(0)2pi((b^(3)-a^(3))/(3))=sigma_(0)2pi(b-a)K^(2)`
`=K=sqrt((a^(2)+b^(2)+ab)/(3))`
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