Final answer:
The formation of a black hole depends on the object's mass being compressed below its Schwarzschild radius, and not on rotation speed. The Schwarzschild radius is calculated from the mass of the object, and once this radius is reached, the object collapses into a singularity from which light cannot escape.
Step-by-step explanation:
The question of how fast an incompressible ball with mass M and radius R must rotate to form a black hole cannot be answered directly with a simple formula for rotational speed because the creation of a black hole is not about rotation speed but about density and the object's radius in relation to the Schwarzschild radius (Rs). For an object to become a black hole, its mass must be compressed to a size equal to or smaller than this critical radius.
The Schwarzschild radius (Rs) is calculated using the formula Rs = 2GM/c², where G is the universal gravitational constant, M is the mass of the body, and c is the speed of light. If an object's radius is compressed below its Schwarzschild radius, it becomes a singularity, and gravity at this point becomes so intense that not even light can escape, creating a black hole.
Now, if we consider the provided information about Sgr A* with a mass of about 4 million solar masses, its Schwarzschild radius can be calculated using Rs = 2 * (6.67 × 10⁻¹¹ N·m²/kg²) * (4 × 10⁶) * (1.99 × 10³° kg) / (3.00 × 10⁸ m/s)² which leads to an Rs of 1.18 × 10¹° meters. This is an example of how mass affects the size of a black hole.