In a vacuum, both the 3x3 foot block of gold and the 1x1 inch block of sulfur would fall towards the Earth's surface with the same acceleration due to gravity. This phenomenon is known as the equivalence principle, which states that the gravitational mass and the inertial mass of an object are equivalent. In simpler terms, it means that the gravitational force experienced by an object is directly proportional to its mass.
The acceleration due to gravity near the Earth's surface is approximately 9.8 meters per second squared (9.8 m/s²). This means that regardless of the mass or composition of an object, it will experience the same acceleration when falling freely under the influence of gravity in a vacuum.
The reason for this can be understood through the universal law of gravitation formulated by Isaac Newton. According to this law, the gravitational force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. Mathematically, this is expressed as F = G * (m1 * m2) / r^2, where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the objects, and r is the distance between their centers.
In the case of the falling objects, the gravitational force acting on each object is given by F = m * g, where m is the mass of the object and g is the acceleration due to gravity. By setting these forces equal to each other, we can see that m1 * g1 = m2 * g2. Since the acceleration due to gravity is the same (g1 = g2), the masses of the objects (m1 and m2) cancel out, resulting in the same acceleration for both objects.
In summary, the reason why objects of different masses fall at the same rate in a vacuum is due to the equivalence principle and the fact that the gravitational force is directly proportional to an object's mass.