A thin rod with length L has a linear density that varies as D = dm / dx = Ax ,

Question

A thin rod with length L has a linear density that varies as D = dm/dx = Ax, where A has units of kg/m2(so the density increases as you go further from the axis).
a) Calculate the total mass of the rod in terms of A and L.
b) Calculate the moment of inertia of the rod for an axis at the left end, perpendicular to the rod. Compare your result to that of a uniform rod.
c) Suppose the rod is flipped around so the denser part is now at the origin. Calculate the new moment of intertia and compare to the value from part b and to a uniform rod.

Explanation / Answer

a) Mass = integral of A*x dx from 0 to L

so M = A[x^2/2] o to L

or, M = AL^2/2

b) I = integral dm*x^2

==> I = integral of A*x^3 dx from 0 to L

or, I = A[x^4/4] from 0 to L

or, I = AL^4/4

c)D = dm/dx = A[L-x]

so I = integral of A[L-x]*x^2 dx from 0 to L

or, I = integral of (ALx^2-Ax^3) dx from o to L

or, I = AL [x^3/3]0 to L - A[x^4/4]0 to L

or, I = AL*L63/3 -AL^4/4

or, I = AL^4/3 - AL^4/4

or, I = AL^4/12

it is less than the I in part B and the I of a uniform rod is ML^2/12

so

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