Thin rod nonuniform linear mass density

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SUMMARY

The discussion focuses on calculating the moment of inertia for a thin rod with a nonuniform linear mass density defined by λ(x) = ae^-bx, where 'a' is the linear density and 'b' is a constant. The moment of inertia is derived using the formula I = ∫dm r², with dm expressed as λ(x)dx and r as the distance from the rotation axis. The participant's solution yields I = -e^(bL)a[L²/b + 2L/b² + 2/b³] + 2a/b³, but concerns about unit consistency arise, particularly regarding the dimensions of 'a' and 'b'.

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Homework Statement



A thin rod of total length L has a nonuniform density given by λ(x)=ae^-bx where x is distance from one end of the rod. What is the moment of inertia of the rod for rotations around the end at x=0. Measuring from x=0 to x=L

Homework Equations


λ(x)=ae^-bx
Inertia=I=∫dmr^2
dm=λ(x)dx
r=x
I=mr^2


The Attempt at a Solution


my answer is -e^(bL)a[L^2/b + 2L/b^2 + 2/b^3] + 2a/b^3

I do not think this is correct because my units are not in mass*length^2

 
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It looks correct. a and b mus have dimensions. a is linear density, [a]=[mass]/[length] and =1/[length].

ehild
 

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