Fill in the blank: The center of mass of a uniform rod with length 𝑙 is at what.
Remember, in a uniform gravity field, the center of mass or center of gravity of an object is the unique point from which the object’s weight force acts. In other words, it allows us to assume that the object’s mass is concentrated at the center of mass as if it were a particle. And if we support a rigid body at its center of mass, then it will be perfectly balanced at this single point. So, what does this mean if we’re thinking about the center of mass of a uniform rod? Well, a uniform rod is a rigid object, and we model them as having a constant density.
Now we know that in a rigid body with constant density, the center of mass is located at the geometric center of the body. So, what does this mean if the rod has a length of 𝑙 length units? Well, the center of mass in this case is at the midpoint of the uniform rod. We can therefore calculate the distance that the center of mass is located from either end of the rod by halving the length itself.
Half of 𝑙 is 𝑙 over two. So, we can fill in the blank using this measurement. The center of mass of a uniform rod with length 𝑙 is at 𝑙 over two. So, our final answer is 𝑙 over two.