Question

A thin stick is placed along the x axis so that one of its ends is at x₁ = 0.00 m and the other one at x₂ = 2.95 m. The stick has a non-uniform mass density λ(x) = ax b, where a = -0.041 kg/m² and b = 1.187 kg/m. What is the coordinate of the center of mass of the stick?

Answer 1

To find the coordinate of the center of mass of a non-uniform density stick, we calculate the total mass by integrating the density over its length. The moment density is then integrated, and this result is divided by the total mass to determine the center of mass coordinate.

Step-by-step explanation:

The student is asking how to find the coordinate of the center of mass of a non-uniform stick placed along the x-axis. The mass density of the stick is given as a function λ(x) = axb, with specific values for constants a and b. To find the center of mass, we need to integrate over the length of the stick, taking into account its varying density.

To solve this, we use the equation for the center of mass of a one-dimensional object:

xCM = (1/M)∫ xλ(x)dx,

where M is the total mass of the stick, found by integrating the mass density over its length:

M = ∫ λ(x)dx,

and xλ(x) is the moment density function. We then integrate xλ(x) over the length of the stick to find the numerator of the center of mass equation.

By carefully executing these integrations from x1 = 0.00 m to x2 = 2.95 m with the given values for a and b, we obtain both M and the integral of xλ(x). Finally, we divide the integrated moment density by the total mass to find the center of mass of the stick, which is the coordinate xCM.