Question

A non-uniform rod has mass m and length l. If one end of the rod is placed at x = 0, the mass per unit length of the rod varies according to ��(x) = b (l - x), where b is a constant to be determined. What is the position of the center of mass of this rod?

Answer 1

To find the position of the center of mass of a non-uniform rod, integrate the mass per unit length over the length of the rod. The position of the center of mass is found to be l/3 from the end of the rod.

Step-by-step explanation:

To find the position of the center of mass of the non-uniform rod, we need to integrate the mass per unit length over the length of the rod. In this case, the mass per unit length is given by ��(x) = b (l - x). To determine the constant b, we can use the fact that the total mass of the rod is m. We integrate the mass per unit length from x = 0 to x = l and set it equal to m:

m = ∫[0, l] b (l - x) dx

Solving this integral, we find that b = 2m/l^2. Now, to find the position of the center of mass, we integrate the product of the mass per unit length and the position x from x = 0 to x = l:

x_cm = (1/m) ∫[0, l] x * b (l - x) dx

After integrating and simplifying, we find that x_cm = l/3. Therefore, the center of mass of the non-uniform rod is located at a distance of l/3 from the end of the rod.