We have this relationship for force per unit mass: a = F/m = GM/r².

This gives the acceleration at a distance r from a mass M. I read that “this formulation shows how a test mass experiences acceleration due to gravity without explicitly including the test mass m itself.” What is the definition of “test mass”? To me test mass sounds like a sophistry to say that a mass exists and exists not as the case may be. Can you explain how we are supposed to believe that even though the mass m is not stated explicitly in the formula, it is still there and interacts with the mass M?

The term “test mass” is not a sophistry, but rather a conceptual tool in physics that helps us understand gravitational interactions.

A “test mass” is a hypothetical object [ you introduce “hypothetical objects” in physics and this is not sophistry? ] with the following properties:

• It has mass small enough that its own gravitational field doesn’t significantly affect the gravitational field it’s measuring [ so it has mass but not really and this is not sophistry ! ]
  
• It can be treated as a point particle that experiences forces but doesn’t disturb the system it’s measuring [ this sounds exactly like sophistry to me. It is neither this or that but at the same time, it is both. ]
  

In the equation \(a = GM/r^2\), the mass \(m\) does appear implicitly: [ what does “appear implicitly mean? Are we supposed to see invisible terms with out miraculous Newtonian vision? ]

1. Starting with Newton’s law of gravitation: \(F = GmM/r^2\)
   
2. Dividing both sides by \(m\): \[\frac{F}{m} = a = \frac{GM}{r^2}\]
   

Your mass \(m\) cancels out mathematically, but it’s still conceptually present. [ How? ] This cancellation shows an important physical principle: all objects, regardless of their mass, experience the same gravitational acceleration in a given gravitational field (ignoring air resistance).

This is why Galileo’s apocryphal experiment dropping objects from the Leaning Tower of Pisa showed that objects of different masses fall at the same rate. The mass \(m\) is still interacting with \(M\); it’s just that the acceleration experienced doesn’t depend on the value of \(m\). [ And this how we indicate this situation, with an invisible \(m\). I need to look at this carefully. ]

The beauty of this formulation is that it lets us describe a gravitational field [ there is no gravitational field ] at any point in space without having to specify what object we might place there - the acceleration any object would experience at that point is determined solely by the mass \(M\) and distance \(r\).

Now we need to explain “point particle” which is also a problematic concept for me. A point particle is a mathematical point which can have mass as needed or not case by case. But if point particle is a mathematical point it cannot have mass. Explain.

I understand your conceptual difficulty with point particles. It does seem paradoxical at first: how can something with zero physical dimensions have mass?

A point particle in physics is indeed a mathematical abstraction. It’s an idealization where we treat an object as if all its mass is concentrated at a single point in space, with zero volume [ so physicists ask us to believe something with zero volume ].

Here’s how physicists reconcile [rationaliz ? ] this apparent contradiction:

1. Modeling simplification: Point particles are mathematical models that simplify physical reality. When the size of an object is much smaller than the distances involved in the problem, treating it as a point greatly simplifies calculations without significantly affecting accuracy. [ yes this is true but physicists define a local object as a point particle ]
   
2. Scale-dependent approximation: Whether treating something as a point particle works depends on the scale of observation. An electron might be treated as a point particle in atomic physics, while in quantum field theory, it has more complex properties.
   
3. Properties without extension: In physics models, we assign physical properties (mass, charge, spin) to mathematical points. This works because the mathematical framework allows us to associate these properties with spatial locations without requiring physical extension.
   
4. Different from physical reality: Physicists recognize that point particles are idealizations. Real particles like electrons might be excitations in quantum fields rather than truly point-like objects.
   

This approach lets us create useful mathematical models even though we know real objects have spatial extension. It’s similar to how we might treat a planet as a point mass when calculating its orbit around a star, even though we know planets are not actually points. [ we can do this because the mass of the planet does not enter the effective formulas but we still talk about mass to save Newton’s sacred authority ]

The usefulness of the concept outweighs its philosophical peculiarity - it’s a tool that gives correct predictions despite its idealized nature. [ or it is an idealization invented to save Newton’s authority in other words calculations will still work without all this Newtonian garbage. ]