Center of Mass (Statics)

Determining the center of mass and center of gravity for discrete and continuous systems.

General idea of the topic

The centre of mass is the point that represents a whole system when its translational motion or its equilibrium is studied: it is the mass-weighted average of the positions. If the system is rigid and the gravitational field uniform, it coincides with the centre of gravity, and an external force applied at it produces no rotation.

The topic defines the concept and computes it in two cases: systems of point masses, by sums, and extended bodies, by integrals with linear, surface or volume density. It also insists that it is a computed point, not always a point where there is matter.

What the student is expected to understand

By the end, the student should be able to compute the centre of mass of a set of point masses, of symmetric bodies and of simple continuous distributions, and to use symmetry to simplify.

The student should also be able to interpret the centre of mass as the point that moves like a particle under the total external force.

Recommended learning route

1. Definition of the Centre of Mass: defines the centre of mass as a weighted average and its relation to equilibrium.
2. Centre of Mass in Discrete Systems: computes it for a finite number of point masses along an axis or in a plane.
3. Centre of Mass in Extended Bodies: extends it to continuous bodies with integrals and densities.

Essential physical criterion

The essential criterion is not to confuse the centre of mass with the geometric centre (centroid): they coincide if the density is uniform, even if the body is not symmetric. Before integrating, it is worth looking for symmetries, which locate the centre of mass without calculation if the mass distribution is symmetric, and checking that the result falls between the extreme positions; it may fall outside the material, as in a ring.

Connection with real problems

• stability of vehicles and resting objects.
• sport: jumps, balances and throws.
• engineering: design of structures and loads.
• motion of systems of particles and rigid bodies.

Final goal of the topic

The final goal is to compute the centre of mass of any discrete or continuous system rigorously and to interpret it as the point that represents its overall motion.