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Find the MI of a rod about an axis throu...

Find the `MI` of a rod about an axis through its centre of mass and perpendicular to the length whose linear density varies as `lamda = ax` where a is a constant and `x` is the position of an element of the rod relative to it's left end. The length of the rod `l`.

Text Solution

Verified by Experts

The correct Answer is:
`al^(4)//36`
`X_(cm) =(int xdm)/(int dm) rArr r_(cm) = (int_(0)^(l) x lamda dx)/(int_(0)^(l) lamda dx)`
`X_(cm) =(int_(0)^(l) xaxdx)/(int_(0)^(l) axdx) rArr x_(cm) = (2l)/(3)`
Moment of Inertia about `AB`
`I_(AB) = int_(0)^(l) dmx^(2) rArr I_(AB) = int_(0)^(2) axdxx^(2)`
`I_(AB) = int_(0)^(l) ax^(3) dx rArr I_(AB) = (al^(4))/(4)`
`I_(AB) = I_(CM) + M((2l)/(3))^(2), (al^(4))/(4) = I_(CM) + M((2l)/(3))^(2)`
`I_(CM) = (al^(4))/(36)`.
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