Linear Mass Density and its Variation

The linear mass density (λ) of a rod is defined as the mass per unit length of the rod. In this case, the linear mass density is given by the equation λ = kx^2, where k is a constant and x is the distance from one end of the rod, denoted as point A.

Finding the Center of Mass

To find the distance of the center of mass (COM) from end A, we need to integrate the linear mass density over the length of the rod and divide it by the total mass of the rod.

Integration
Integrating the linear mass density λ = kx^2 over the length of the rod, we have:

M = ∫λdx = ∫kx^2dx,

where M is the total mass of the rod.

Calculating the Total Mass
To calculate the total mass of the rod, we integrate the linear mass density over the entire length of the rod, from x = 0 to x = L (the length of the rod):

M = ∫(kx^2)dx from 0 to L.

Integrating this expression, we get:

M = (k/3)L^3.

Calculating the Position of the Center of Mass
To find the position of the center of mass, we divide the integral of x times the linear mass density by the total mass of the rod:

xCOM = ∫(xλ)dx / M.

Substituting the given expression for λ = kx^2, we have:

xCOM = ∫(kx^3)dx / M.

Integrating this expression, we get:

xCOM = (k/4)L^4 / M.

Substituting the expression for M, we have:

xCOM = (k/4)L^4 / ((k/3)L^3),

Simplifying further, we get:

xCOM = 3L/4.

Conclusion
Therefore, the distance of the center of mass from end A is 3L/4. This means that the center of mass is located closer to the opposite end of the rod, at a distance of 3/4 times the total length of the rod.

COM
x=3L/4