Particle Motion

We are given a rod of length , with a linear mass density that varies along the length of the rod. The linear mass density at a point at a distance from the left end of the rod is given by: The center of mass of the rod can be found using the following formula: First, we calculate the mass of the rod using the linear mass density. The total mass of the rod is given by the integral of over the length of the rod: Performing the integration: Thus, the total mass of the rod is . Now, we calculate the moment of mass distribution about the left end of the rod: Performing the integration: Thus, the integral becomes: Finally, the center of mass is: Thus, the center of mass of the rod is at a distance from the left end. So the correct answer is (C) .