Prove acceleration due to gravity does not depend on the mass of the object mathematically and theoretically .
The short answer
The greater mass is accelerated toward Earth by an proportionately greater influence of gravity. So, although it is harder to change its speed, it is being pulled in with more force.
The theoretical answer
First a question
Why do a hammer and a feather dropped from a height land at the same time?
Equal force yields equal acceleration per unit mass
In your past experience, a hammer feels heavy and so you realize that there is a strong force pulling it down toward the ground. You feel this force as your muscles tighten to hold up the hammer. Your arm must exert a counter force to prevent the hammer from dropping. On the other hand, a feather feels very light. Although you can see it, it is much lighter than your hand, and so you don't sense any downward force.
So far, we have a general feeling that the force of gravity (at least, at a fixed distance from earth) is proportional to (or at least related to) the mass of an object. In fact, gravity exerts a force that is equal on every unit mass (for example, it is equal for each gram of mass).
So, the real question is: If gravity applies 10,000 times more force on the hammer, why does it not fall faster? The answer is surprisingly simple. The hammer is so massive that it is much harder to accelerate. In fact, it requires 10,000 times the force to accelerate at the same rate as an object that is 10k less massive.
Conclusion:
Gravitational force and acceleration in the presence of an external force are both proportional to an object’s mass. (According to Newton’s law of motion, acceleration is inversely proportional to mass.). ‘Light’ objects experience less force, but are easy to move. Massive objects experience greater force, but are hard to move. For this reason, in the absence of air friction, objects of any mass accelerate toward a large mass at the same rate. They also orbit with the same period. That is, the duration of a single orbit is related to the distance from the gravitational center, but not the mass of the object.
Mathematically
The acceleration due to gravity never depends on the mass.
The acceleration of mass m due to mass M is given by:
F=GMm/r2=ma⟹a=GM/r2F=GMm/r2=ma⟹a=GM/r2
… see, does not depend on m (does depend on M though … that is, it depends on the other mass!)
This makes sense, after all: all macriscopic objects are made from an assortment of bits of different masses. If different masses fell at different rates, then the object will have to decide which mass to use to dictate it’s overall rate of falling.
The trouble with this comes when you have two masses falling next to each other, attached by a string. The falling rate would have to be different if the string is cut. Does this make sense?
The short answer
The greater mass is accelerated toward Earth by an proportionately greater influence of gravity. So, although it is harder to change its speed, it is being pulled in with more force.
The theoretical answer
First a question
Why do a hammer and a feather dropped from a height land at the same time?
Equal force yields equal acceleration per unit mass
In your past experience, a hammer feels heavy and so you realize that there is a strong force pulling it down toward the ground. You feel this force as your muscles tighten to hold up the hammer. Your arm must exert a counter force to prevent the hammer from dropping. On the other hand, a feather feels very light. Although you can see it, it is much lighter than your hand, and so you don't sense any downward force.
So far, we have a general feeling that the force of gravity (at least, at a fixed distance from earth) is proportional to (or at least related to) the mass of an object. In fact, gravity exerts a force that is equal on every unit mass (for example, it is equal for each gram of mass).
So, the real question is: If gravity applies 10,000 times more force on the hammer, why does it not fall faster? The answer is surprisingly simple. The hammer is so massive that it is much harder to accelerate. In fact, it requires 10,000 times the force to accelerate at the same rate as an object that is 10k less massive.
Conclusion:
Gravitational force and acceleration in the presence of an external force are both proportional to an object’s mass. (According to Newton’s law of motion, acceleration is inversely proportional to mass.). ‘Light’ objects experience less force, but are easy to move. Massive objects experience greater force, but are hard to move. For this reason, in the absence of air friction, objects of any mass accelerate toward a large mass at the same rate. They also orbit with the same period. That is, the duration of a single orbit is related to the distance from the gravitational center, but not the mass of the object.
Mathematically
The acceleration due to gravity never depends on the mass.
The acceleration of mass m due to mass M is given by:
F=GMm/r2=ma⟹a=GM/r2F=GMm/r2=ma⟹a=GM/r2
… see, does not depend on m (does depend on M though … that is, it depends on the other mass!)
This makes sense, after all: all macriscopic objects are made from an assortment of bits of different masses. If different masses fell at different rates, then the object will have to decide which mass to use to dictate it’s overall rate of falling.
The trouble with this comes when you have two masses falling next to each other, attached by a string. The falling rate would have to be different if the string is cut. Does this make sense?
