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The mass per unit length (lambda) of a n...

The mass per unit length `(lambda)` of a non-uniform rod varies linearly with distance x from its one end accrding to the relation, `lambda = alpha x`, where `alpha` is a constant. Find the centre of mass as a fraction of its length L.

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In this case, we replace dm by `lambda dx`, where `lambda` is not constant. Therefore,`x_(cm)` is `x_(cm)=1/Mint_(0)^(L)lamdadm=1/Mint_(0)^(L)xlamdadx=alpha/Mint_(0)^(L)x^2dx=(lamdaL^3)/(3M)`We can eliminate a by noting that the total mass of the rod is elated to a through the relationship `M=int dm=int_(0)^(L)lamdadx=int_(0)^(L)alphaxdx=(alphaL^3)/2`Substituting this into the expression for `x(CM)` gives `x_(cm)=(alphaL^3)/((3alphaL^2 )/2)=2/3L=4m`.
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