Does the Acceleration Due to Gravity Depend on Mass, Weight or Both?

The question of whether the acceleration due to gravity depends on mass, weight, or both is a fundamental inquiry in physics. This article aims to clarify the relationship between gravity, mass, and weight through an exploration of Newton's laws of motion and gravitational force.

Understanding Gravity and Newton's Law of Universal Gravitation

According to Newton's law of universal gravitation, the gravitational force between two masses is given by the equation:

F G x m1 x m2 / r2

Where:

F is the force of gravity between the two masses, G is the gravitational constant, m1 and m2 are the masses of the two objects, r is the distance between the centers of the two masses.

This law defines the gravitational force between masses, but what about the acceleration due to gravity? According to Newton's second law of motion, force is also defined as:

F m x a

Where:

F is the force, m is the mass of the object, a is the acceleration.

When considering the acceleration due to gravity (g), the equation becomes:

Fg m x g

Since the force of gravity between an object and the Earth is also given by:

Fg G x m x M / r2

Where:

M is the mass of the Earth, r is the distance from the object to the center of the Earth.

We can equate the two expressions for gravitational force:

Fg m x g G x m x M / r2

Solving for g, we get:

g G x M / r2

This equation shows that the acceleration due to gravity depends on the mass of the Earth (M) and the distance from the object to the center of the Earth (r). Therefore, it does not depend on the mass or weight of the falling object itself.

Acceleration and the Effect of Mass

The key point to remember is that near the surface of the Earth, in the absence of air resistance, all objects accelerate at the same rate, regardless of their mass or density. This constant acceleration is approximately 9.8 m/s2. The equation for this is derived from Newton's second law:

F m x a

In the case of acceleration due to gravity:

Fg m x g

Since the force of gravity acting on the object (Fg) is given by:

Fg G x m x M / r2

Substituting Fg into the acceleration equation, we get:

m x g G x m x M / r2

Solving for g:

g G x M / r2

As can be seen, the acceleration due to gravity is independent of the mass of the falling object. It only depends on the mass of the Earth and the distance from the object to the Earth's center.

Air Resistance and Density Effects

However, in real-world scenarios, air resistance, buoyancy, and other factors can influence the acceleration of a falling object. Air resistance acts in the opposite direction to the motion and depends on the object's shape and surface area. Buoyancy is significant when an object is falling through a medium with a different density, such as water compared to air.

For example, consider a large solid aluminum object dropped into water. Its buoyancy must be taken into account. The upward buoyant force (F_b) is equal to the weight of the water displaced by the object. The net force acting on the object becomes:

Fnet Fg - Fb

Where:

Fg is the gravitational force (weight) of the aluminum object, Fb is the buoyant force.

The acceleration of the object in water is thus:

a Fnet / m (Fg - Fb) / m

This acceleration (a) will be less than 9.8 m/s2 due to the reduction in net force. The density of the object and the medium affect this calculation.

In summary, in the absence of air resistance or other significant density differences, the acceleration due to gravity is independent of the mass and density of the falling object. The acceleration depends only on the mass of the Earth and the distance from the object to the Earth's center.