A slender rod with length L has a mass per unit length that waries with distance

Question

A slender rod with length L has a mass per unit length that waries with distance from the left hand end, where x = 0, according to dm/dx = gama x, where gama has units of kg/m^2. Calculate the total mass of the rod in terms of gama and L. Express your answer in terms of the variables gama and L. Use I = integral r^2 dm to calculate the moment of inertia of the rod for an ands at the left-hand and, perpendicular to the rod. Use the expression you derived in to express I in terms of M and L. Express your answer in terms of the variables M and L. Report part B an axis at the right-hand and of the rod. Express your answer in terms of the variables M and L.

Explanation / Answer

dM = (r*x)*dx = r*x*dx

M = r*x^2/2 = r*L^2/2

+++++++++++++++

dI = dm*x^2 = (r*x)*dx*x^2

dI = r*x^3*dx

I = integration dI

I = r*x^4/4

I = r*L^4/4 = M*L^2/2

++++++++++++++++++

dI = dm*(L-x)^2 = (r*x)*dx*(L-x)^2

dI = r*L*x*dx - r*x^3*dx

I = integration dI from x = 0 to L

I = r*L*x^2/2 - r*x^4/4

I = r*L^3/4 - r*L^4/4

I = ML/2 - ML^2/2

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