The centripetal force is given by:

\[ F = \frac{mv^2}{r} \]

Here, \( F \) is the force needed to keep the body moving in a circular path. Dividing \( F \) by the mass \( m \), we get the centripetal acceleration:

\[ a = \frac{F}{m} = \frac{v^2}{r} \]

Notice that \( a \) depends only on the velocity \( v \) of the body and the radius \( r \) of the circular motion, but not on the mass \( m \).

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The “centrifugal tendency” you refer to is likely the inertial resistance of the body to the centripetal force. According to Newton’s first law of motion, a body in motion tends to remain in motion in a straight line unless acted upon by an external force. In circular motion, the body “wants” to move tangentially due to inertia. This resistance or tendency is reflected in the requirement of a centripetal force to maintain the circular motion.

Since the centripetal acceleration \( a = v^2/r \) is independent of the mass \( m \), the inertial tendency to “move outward” (what we might call the centrifugal tendency in the non-inertial frame) is also independent of \( m \).

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This independence from mass in circular motion is conceptually similar to the observation that gravitational acceleration \( g \) is the same for all objects near Earth’s surface, regardless of their mass. In both cases, the acceleration is determined by factors other than the mass of the body itself:

• For centripetal motion, \( v \) and \( r \) dictate \( a \).
  
• For free-fall motion, \( g \) depends on the gravitational field.
  

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The independence of the centrifugal tendency from the mass of the body highlights a key feature of Newtonian mechanics: the universality of acceleration in the context of forces. This universality underlies much of classical physics and aligns with principles such as the equivalence of inertial and gravitational mass.

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Yes, you can assume that the centrifugal tendency (or the inertial resistance to circular motion) is independent of the mass of the body. This reflects the fact that the centripetal acceleration required for circular motion depends only on the velocity and radius of the motion, not on the mass of the object in motion.

The orbital period \( T \) is the time it takes for one complete revolution along the circular path. The distance traveled in one revolution is the circumference of the circle, \( 2\pi r \), where \( r \) is the radius of the circular path.

The velocity \( v \) is then given by:

\[ v = \frac{\text{Distance traveled in one period}}{\text{Time for one period}} = \frac{2\pi r}{T} \]

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The angular velocity \( \omega \) is the rate at which the object sweeps out angles in radians per second. It is related to the period \( T \) by:

\[ \omega = \frac{2\pi}{T} \]

The tangential velocity \( v \) is related to \( \omega \) by the radius \( r \):

\[ v = \omega r \]

Thus, using \( T \), we can rewrite \( v \) as:

\[ v = \left(\frac{2\pi}{T}\right)r = \frac{2\pi r}{T} \]

This confirms the equivalence of the two methods.

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• From \( T \): \( v = \frac{2\pi r}{T} \)
  
• From \( \omega \): \( v = \omega r \), where \( \omega = \frac{2\pi}{T} \)
  

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1. In many practical cases, you can measure the distance an object travels in a given time (e.g., track the motion of a planet or a car on a circular track), giving \( v \) directly.
   
2. Orbital period \( T \) often requires observing a full revolution, which can take a long time (e.g., for planets).
   
3. While angular velocity \( \omega \) is useful in rotational mechanics, measuring tangential velocity \( v \) directly (e.g., using speed sensors) is more common in real-world applications.
   

By using \( v \), you bypass the need to measure \( T \) or \( \omega \) explicitly, making it a more intuitive and practical starting point for many problems in circular motion.

• Definition: Centripetal force is the force that acts on a body moving in a circular path, directed toward the center of the circle. It keeps the object in its curved trajectory.
  
• Formula:
  \[ F_{\text{centripetal}} = \frac{mv^2}{r} \]
  

This force is required to counteract the body’s natural tendency (due to inertia) to move in a straight line, as stated by Newton’s First Law.

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• Inertia is the property of a body to resist changes in its state of motion. If no net force acts on the body, it will continue moving in a straight line at constant speed.
  
• In circular motion, an object’s inertia causes it to “want” to move in a straight line tangential to the circle at every instant.
  
• To keep the object in circular motion, a centripetal force must act inward, toward the center of the circle. Without this force, the object would fly off tangentially.
  

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• The “centrifugal tendency” refers to the apparent outward effect observed from the perspective of someone in the rotating frame.
  
• This “outward” effect is not a real force but an inertial effect. It’s the result of the body’s resistance (inertia) to the inward centripetal force. Essentially, the body “pushes back” against the centripetal force due to its inertia.
  

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In Newtonian mechanics:

• The centripetal force is a real, physical force that pulls the body inward.
  
• The body’s reaction, due to its inertia, is what creates the sensation of being “pushed outward” in a rotating reference frame.
  

This “reaction” is often referred to as the centrifugal tendency, although it’s not a separate force in the inertial frame of reference. In a rotating (non-inertial) frame, this effect is modeled as the centrifugal force, which is a fictitious or apparent force.

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When I said:
> (and consequently the “centrifugal tendency,” if we consider the inertial reaction to centripetal force)

I meant:

• The body’s tendency to resist being accelerated inward (due to inertia) gives rise to the sensation of being “pushed outward.”
  
• This is the **centrifugal tendency**—an inertial effect caused by the body’s natural resistance to changes in its straight-line motion.
  

This tendency is independent of the mass \( m \) because inertia affects all objects equally, as described earlier. The sensation of being “pushed outward” (what we call “centrifugal force” in a rotating frame) is a direct consequence of the inward centripetal force acting on the body.

Can you translate this sentence from Turkish to English and comment what you think about it in terms of physics or philosophy or even history of science: “Doğal hareket dairesel hareketse ve hareket sadece transfer edilebiliyorsa, bütün hareketlerin dairesel olması gerekir.”

Translation:

“If natural motion is circular motion and if motion can only be transferred, then all motions must be circular.”

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Commentary:

This sentence sets up a logical argument based on two premises:

1. Natural Motion Is Circular:
   The claim that natural motion is inherently circular has deep roots in classical and medieval cosmologies. For example, in Aristotelian physics, the heavens were thought to be perfect and eternal, moving in perfect circles because the circle was considered the most perfect geometric form. This idea was influential in shaping early scientific thought about celestial mechanics.
   
2. Motion Can Only Be Transferred:
   The notion that motion cannot be created or destroyed but only transferred resembles an early intuition about conservation principles. While not a full formulation of the modern conservation of momentum or energy, it hints at the idea that motion must come from somewhere—it is passed from one body to another.
   

Implications in Physics and Philosophy:

• Physics:
  If both premises were true, then every motion in the universe would be circular. However, modern Newtonian mechanics shows us that in the absence of an external force, an object will continue in a straight line (Newton’s First Law of Inertia). Circular motion, in fact, requires a centripetal force acting on an object. Thus, the conclusion that “all motions must be circular” contradicts our empirical observations of linear and other forms of motion. This discrepancy highlights the evolution from early ideas about motion to the more accurate descriptions provided by classical mechanics and, later, modern physics.
  
• Philosophy:
  Philosophically, the sentence is a good example of reasoning based on foundational assumptions. It shows how starting from certain premises (here, the nature of motion and the means by which it is transferred) can lead to broad conclusions that must be checked against reality. The absurdity of the conclusion (since we observe many non-circular motions) suggests that one or both of the initial assumptions must be reconsidered. This kind of reasoning—deriving consequences from basic premises and then testing them against experience—is at the heart of the scientific method.
  
• History of Science:
  Historically, the debate over the nature of motion was central to the transition from Aristotelian physics to the modern scientific worldview. The Aristotelian view, with its division between the “natural” (often circular in the heavens) and “violent” (often linear on Earth) motions, was gradually replaced by the unified framework of inertia and gravitational force articulated by Galileo and Newton. The argument encapsulated in the sentence reflects an earlier stage in this evolution, where the inherent properties of motion were still being debated and where logical deductions—if taken at face value—could lead to conclusions that we now know to be false.
  

In summary, while the sentence may seem abstract, it captures a pivotal moment in the history of scientific thought: an era when the assumptions about motion were being rigorously scrutinized. It reminds us that the path to our current understanding of the physical world has been paved with hypotheses and logical arguments—some of which led to dead ends—that ultimately forced us to refine our ideas to match observed reality.