Question

A thin stick is placed along the x axis so that one of its ends is at x₁ = 0.0m and the other one at x₂ = 3.95m. the stick has a non-unfirom mass density (x) = ax b, where a = 0.089 kg/m² and b = 1.387 kg/m. what is the x coordinate xcm of the stick's center of mass?

Answer 1

To find the x-coordinate of the center of mass (xcm) of the stick, we need to calculate the average position of all the mass elements along the x-axis using integration. The x-coordinate of the center of mass can be found using the equation xcm = (1/M) * a * [(x^(b+2))/2] + C, where a and b are given values. Substituting the values of a = 0.089 kg/m² and b = 1.387 kg/m, we can calculate the xcm.

Step-by-step explanation:

To find the x-coordinate of the center of mass (xcm) of the stick, we need to calculate the average position of all the mass elements along the x-axis. Since the mass density of the stick is non-uniform and given by p(x) = ax^b, we can treat each small element of the stick as a point mass at a specific position.

We can divide the stick into small segments of length dm and treat each segment as a point mass. The mass of each small segment is dm = p(x) * dx = a * x^b * dx. The total mass of the stick is M, so we need to integrate the mass elements over the length of the stick:

xcm = (1/M) * ∫x * dm = (1/M) * ∫(x * a * x^b * dx) = (1/M) * a * ∫(x^(b+1)) * dx = (a/M) * [∫(x^(b+2))/2] + C

Using the given values, a = 0.089 kg/m² and b = 1.387 kg/m, we can substitute them into the equation to find the x-coordinate of the center of mass.