Introduction

When three identical spheres, each of radius R, are placed on a horizontal surface and touching one another, the system is in a balanced state. The center of mass of the system is at the point where the three spheres touch. However, if one of the spheres is removed, the center of mass of the system will shift. This shift can be explained by considering the change in distribution of mass in the system.

Explanation

1. Initial state:
In the initial state, the three spheres are placed in contact with each other, forming a stable configuration. The center of mass of the system is at the point where the spheres touch. The distribution of mass is symmetric, with equal mass on either side of the center of mass.

2. Removing a sphere:
When one of the spheres is removed, the distribution of mass in the system changes. The remaining two spheres will still be in contact with each other, but the center of mass will shift.

3. New center of mass:
The center of mass of the system will now be closer to the two remaining spheres. This can be explained by considering the distribution of mass. Since the removed sphere had mass, its absence results in a decrease in the total mass of the system. As a result, the center of mass shifts towards the remaining spheres, which still contribute to the mass of the system.

4. Shift direction:
The direction of the shift in the center of mass depends on the relative positions of the remaining spheres. If the two remaining spheres are equidistant from the removed sphere, the center of mass will shift towards the midpoint between them. However, if the remaining spheres are not equidistant, the center of mass will shift towards the sphere that is closer to the removed sphere.

5. Magnitude of shift:
The magnitude of the shift in the center of mass depends on the mass of the removed sphere compared to the mass of the remaining spheres. If the removed sphere had a significant mass, the shift will be more pronounced. On the other hand, if the removed sphere had a negligible mass compared to the remaining spheres, the shift will be minimal.

Conclusion

When one of the three identical spheres, each of radius R, is removed from a configuration where they are touching each other on a horizontal surface, the center of mass of the system will shift. This shift occurs due to the change in mass distribution in the system, resulting in the center of mass moving closer to the remaining spheres. The direction and magnitude of the shift depend on the relative positions and masses of the spheres.

R/root3 is the answer