Two air track gliders are connected by a spring, in the middle of which is a red marker. With the air blower on, set the pair of gliders at one end of the air track, and give the lead glider a gentle push down the track. As the pair of gliders moves down the track, the gliders oscillate about the marker, while the marker travels with constant velocity.
The center of mass of an object is the weighted mean displacement of all of its mass points from some reference point, usually the origin of the coordinate system in which the object sits. This is the sum of the products of all the mass points with their distances from the origin, divided by the total mass of the object (rcm = Σmiri/M). It is usually most convenient to set the center of mass of the object at the origin, but whatever our reference point, for the pair of gliders connected by a spring, shown in the photograph above, it is at the location marked by the red marker attached to the middle of the spring.
Whether a system is in motion or at rest, unless an external force acts upon it, its center of mass must either keep moving as it is or remain at rest. The pair of gliders connected by a spring comprise a mass-spring oscillator. If you set the pair of gliders on the air track, pull them apart and then release them without applying any net horizontal force to the pair, they oscillate in place. The red marker stays fixed while the two gliders move back and forth in opposite directions about it. The center of mass of the system remains where it is.
Now set the pair of gliders at one end of the track and give the glider farther from the end a gentle push, so as to stretch the spring slightly and send the pair moving down the track. (You can also give the glider nearer the end of the track a gentle push.) In doing this, you exert a net horizontal force on the pair of gliders, and thus its center of mass, and you also set the pair oscillating on the spring. As the pair of gliders travels along the track, the red marker moves at constant velocity, while the gliders oscillate about it. That is, the center of mass of the pair of gliders connected by the spring, moves at constant velocity, while the gliders oscillate about it.