Gravitational Potential Energy for Extended Objects

Consider a system comprising several point masses. The coordinates of the center of mass for this system can be expressed as the summation of the product of each mass and its position vector divided by the total mass:

Suppose the point masses are replaced with an extended object with uniformly distributed mass. The coordinates of the center of mass for this object can be obtained by replacing the point mass with the differential mass element and the summation with an integral in the equation for the center of mass:

Consider a ring with uniform mass distribution M and radius R. The circular symmetry ensures that the center of mass is located at the ring's geometric center:

Consider a coordinate system with its origin located at the center of the ring. Since the ring has a uniform mass distribution, the linear mass density is constant. So, the differential mass element on the surface of the ring is the product of the linear mass density and the differential length element on the ring's surface.

Now, using the expression for the center of mass and substituting the value of the position vector in the component form and the differential mass element gives the following equation:

As the arc length ds subtends a differential angle dθ, the arc length equals the radius multiplied by the differential angle. The linear mass density is the total mass divided by the length of the ring. Incorporating these values of arc length and linear mass density, the center of mass expression reduces to the following:

The variable of integration is the angle θ. So, the limits of integration around the ring are θ = 0 to θ = 2π. The integral is separated into the x and y components and integrated across the limits:

Since the origin of the coordinate is located at the center of the ring, the center of mass of the ring lies at its geometric center.