# Falling Objects With Different Mass

• BL4CKCR4Y0NS
In summary, Galileo's experiment at the Leaning Tower of Pisa proved that all objects, regardless of their mass, fall at the same rate due to the Earth's gravitational field being uniform. This is because the force of gravity and the mass of the object cancel out in the equation for acceleration, allowing all objects to reach the ground at the same time. While increasing the mass of an object does increase the force of gravity, it also shifts the center of mass closer to the object, keeping its acceleration the same in relation to the center of mass.

#### BL4CKCR4Y0NS

I've heard and read many times about Galileo and his standing up in the Leaning Tower of Piza. How he dropped two objects of different mass and proving that the two objects hit the floor at the same time if released at the same time.

What I never understood ... was WHY this happened. What causes the two objects to hit the floor at the same moment.

This is because, ignoring air resistance, all objects accelerate downwards with the same acceleration, g. While the gravitational force acting on a more massive object is greater than that acting on a less massive object, the more massive object still accelerates downwards at the same acceleration as the less massive object due to the more massive object possessing a larger inertial mass.

To simplify issues, we approximate the Earth's gravitational field to be uniform such that the value of g remains constant. The gravitational force acting on a body of mass m on Earth is then $$F_{g} = mg$$. From the simplified version of Newton's Second Law, we have $$F = ma$$. Combining, we see that the acceleration a of the body is simply g - independent of its mass.

Oh yeah .. that makes sense.

Thanks. :)

Saying that "we approximate the Earth's gravitational field to be uniform such that the value of g remains constant" may appear to be begging the question.

In more detail, The gravitational force between two bodies is $F= GmM/r^2$ where G is a universal constant, m and M are the masses of the two bodies and r is the distance between their centers. Take the Earth to be one body and the object to be dropped to be the other. The radii of two different objects are so small compared to the radius of Earth that "r" can be taken as a constant for any object dropped at the surface of the earth. As Fightfish said, the second law is F= ma so for any object we have $GmM/r^2= ma$ and can cancel the two "m"s: $a= GM/r^2$ for any object dropped at the surface of the earth.

Since two objects always have the same acceleration and start with speed 0 ("dropped" not "thrown down"), the always have the same speed at any time and so always drop the same distance in the same time.

The problem I had in understanding the universality of free fall was this: If increasing the mass of the falling object also increases the force, then why is there no effect on acceleration? The truth of the matter is that the relative acceleration does increase. In other words, the time that it will take for the two objects to meet will decrease. However, the universality of free fall does not refer to relative acceleration because the relative acceleration is a non-inertial frame of reference. The frame of reference for the universality of free fall is the center of mass. So if you increase the mass of the falling object you are also shifting the center of mass closer to the falling object. This has the effect of keeping it's acceleration (relative to the center of mass) the same, even though the relative acceleration has increased. So $a=GM/r^2$ and $a_{rel}=G(M+m)/r^2$.

## 1. What is the relationship between mass and the rate of falling objects?

The rate at which an object falls is determined by its mass and the force of gravity. According to Newton's Second Law of Motion, the acceleration of an object is directly proportional to the net force applied to it and inversely proportional to its mass. This means that objects with a greater mass will fall faster than objects with a smaller mass, assuming they are dropped from the same height and experience the same force of gravity.

## 2. How does air resistance affect the falling rate of objects with different mass?

Air resistance, also known as drag, is a force that acts in the opposite direction of an object's motion through the air. As objects fall, they encounter air resistance which slows down their rate of falling. This means that objects with a larger surface area, such as a feather, will experience more air resistance and fall slower than objects with a smaller surface area, such as a bowling ball.

## 3. What is the difference between mass and weight in relation to falling objects?

Mass and weight are often used interchangeably, but they are not the same. Mass is a measure of the amount of matter in an object, while weight is a measure of the force of gravity on an object. When an object is falling, its mass remains constant, but its weight changes due to the acceleration of gravity. This is why objects of different mass can have the same rate of falling, as long as they experience the same force of gravity.

## 4. Can objects with different mass reach the ground at the same time?

Yes, objects with different mass can reach the ground at the same time if they are dropped from the same height and experience the same force of gravity. This is known as the "Principle of Equivalence" and was famously demonstrated by Galileo in the 16th century by dropping two objects of different mass from the Leaning Tower of Pisa.

## 5. How does the shape of an object affect its rate of falling compared to its mass?

The shape of an object can greatly affect its rate of falling, even if it has the same mass as another object. Objects with a larger surface area, such as a parachute, will experience more air resistance and fall slower than objects with a smaller surface area, such as a pencil. This is because the air resistance is proportional to the surface area of the object, not its mass.