If you have a uniform rod and you hang it from the center, where is the center of mass of the rod
The correct answer and explanation is:
Answer:
If you have a uniform rod and you hang it from the center, the center of mass of the rod is exactly at the center point from which it is hung.
Explanation:
The center of mass (COM) of an object is the point at which the entire mass of the object can be considered to be concentrated for the purpose of analyzing translational motion. For uniform objects, the center of mass coincides with the geometric center because the mass is evenly distributed.
A uniform rod means that the rod’s mass per unit length is constant. Imagine the rod as a straight line segment of length LL and total mass MM. Since the mass distribution is uniform, the mass is spread evenly along the length of the rod.
Mathematically, if you set the origin of your coordinate system at one end of the rod (say, at x=0x = 0) and the other end at x=Lx = L, the center of mass xcmx_{cm} is given by: xcm=1M∫0Lx dmx_{cm} = \frac{1}{M} \int_0^L x \, dm
Because the mass distribution is uniform, dm=MLdxdm = \frac{M}{L} dx, so: xcm=1M∫0LxMLdx=1L∫0Lx dx=1L[x22]0L=L2x_{cm} = \frac{1}{M} \int_0^L x \frac{M}{L} dx = \frac{1}{L} \int_0^L x \, dx = \frac{1}{L} \left[ \frac{x^2}{2} \right]_0^L = \frac{L}{2}
This means the center of mass is at the midpoint of the rod, at L/2L/2.
When you hang the rod from the center, the pivot point coincides with the center of mass. This is why the rod remains balanced and does not rotate about that point—there is no net torque due to gravity acting on the rod, because the weight effectively acts through the pivot.
This principle is fundamental in mechanics and explains why uniform beams or rods balance at their midpoints. If the rod were non-uniform (with mass distributed unevenly), the center of mass would shift toward the heavier side, and the rod would not balance at its geometric center.