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A thin circular plate of mass M and radi...

A thin circular plate of mass M and radius R has its density varying as `rho(r)=rho_(0)r` with `rho_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 = aMR^(2)` The value of the coefficient a is :

A

`1/2`

B

`3/2`

C

`8/5`

D

`3/5`

Text Solution

Verified by Experts

The correct Answer is:
C
`I_(CM) = int dn r^2 = int_(0)^(R ) ( 2pi dr) (rho_(0)r) r^2 = (2pi rho_0R^5)/(5)`
`M = int dn = int_(0)^(R ) 2pi d r rho_0 r = (2pi rho_0R^3)/(3)`
Using parallel axis theorem `I = I_(CM) + MR^(2)`
`= 2/5 pi rho_(0) R^5 + 2/3 pi rho_0 R^5 = 16/15 pi rho_0 R^5`
`=3/2 xx 16/15 xx 2/3 pi rho_(0)R^5 = 8/5 MR^2`.
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