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A rod of length L and cross section area...

A rod of length L and cross section area A has variable density according to the relation `rho (x)=rho_(0)+kx` for `0 le x le L/2` and `rho(x)=2x^(2)` for `L/2 le x le L` where `rho_(0)` and k are constants. Find the mass of the rod.

Text Solution

Verified by Experts

The correct Answer is:
`((7L^(3))/12+rho_(0)L/2+(kL^(2))/8)A`
`M_(1)=underset(0)overset(L//2)(int)A(rho_(0)+kx)dx=(rho_(0) L/2+(kL^(2))/8)A`
`M_(2)=underset(L//2)overset(L)(int)A(2x^(2)dx)=2/3[L^(3)-L^(3)/8]=(14L^(3))/24 A`
`M_("total")=M_(1)+M_(2)=((14L^(3))/24+rho_(0)L/2+(kL^(2))/8)A`
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