k g / m 2 A slender rod with length L has a mass per unit length that varies wit

Question

kg/m 2

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

Explanation / Answer

Total mass = integral (dm) = int(r*xdx)]x=0 to L

= rL^2 /2

where 'r' is 'gamma'

(b) Consider a small section of the rod of length dx, at a distance x from the axis.

mass of the section, dM = rxdx

So, moment of inertia of that section = x^2 * dM = r x^3 dx

Integrating it from x=0 to L,

I = r * x^4 /4 ] x=0,L

I = r*L^4 /4 - 0

= 1/2 * ML^2

(c) Consider a small section of the rod of length dx, at a distance x from the right axis.

mass of the section, dM = r(1-x)dx

So, moment of inertia of that section = x^2 * dM = r x^2 *(L-x) dx

= r*(Lx^2 - x^3) dx

Integrating it from x=0 to L,

I = r * (Lx^3 /3 - x^4 /4) ] x=0,L

I = r*L^4 /12 - 0

= 1/6 * ML^2

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