Understanding Inert Bodies: Newton and Archimedes' Equations

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In summary, Newton and Archimedes both discuss the principles of force and mass, but their approaches differ in terms of time-dependence. Newton's equation focuses on the relationship between acceleration, force, and mass, while Archimedes' equation deals with the equilibrium of forces and distances from the center. Additionally, Newton's calculus is used to analyze dynamic forces, while Archimedes' calculus is used for static forces. Ultimately, understanding the differences between the two can lead to a better understanding of concepts like electric charge and gauge symmetry.
  • #1
deda
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Newton says: “A body accelerates proportionally with the force it carries and reciprocally to its mass”. His equation is:
[tex] \frac {F}{m} = \frac {d^2x}{dt^2}[/tex]
So, what Newton does is consider the position changing while time goes by and while the force and mass remain constant. Conclusively, if the stone is lighter I can throw it further away. This way the mass becomes criterion for how inert one body is. But if the lighter and the heavier stone throw each other then they will do it with equal and opposite forces. So:
[tex] F_1 = -F_2 <=> \frac {M_1}{M_2} = - \frac {a_1}{a_2}[/tex]
[tex]\frac {F_1}{M_1} <> \frac {F_2}{M_2}<=> a_1 <> a_2[/tex]

Archimedes says: “Magnitudes are in equilibrium on reciprocally proportional distances from the center”. His equation is:
[tex]\frac {F_1}{F_2} = \frac {D_2}{D_1} = \frac {M_1}{M_2}[/tex]
And if the forces and masses are again constant then:
[tex]\frac {F_1}{F_2} = \frac {dD_2}{dD_1} = \frac {M_1}{M_2}[/tex]
So, the heavier body will carry more force and will pass smaller distance. Actually the fact that the mass is criterion for how inert one body is comes from Archimedes. Among the differences between the two physics, Archimedes’s one is time - independent.

I tell you: Our physics starts with Archimedes and that physics takes wrong direction when little Newton comes up with his mirage. We cannot blame the kids because they tell the truth even when they lie, yet we cannot consider them seriously.
 
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  • #2
The key to fully understanding the different principles working behind Archimedes' level and Newton's mechanics is to know the difference between statics of force and dynamics of force.

Statics deals with the equilibrium of forces. This is time independent. Intrinsically, the concept of mass must be described by this method of force equilibrium. The math is the integral calculus which by no coincidence is discovered by Archimedes before it was attributed to Leibniz. This is the quantum nature of the Hamiltonian function.

Dynamics deals with the nonequilibrium of forces. This is time dependent. Intrinsically, the concept of electric charge must be described by this method of force nonequilibrium. The math is the differential calculus and invented by Newton to deal with this method of analysis. This is the beginning of the Lagrangian function and gauge symmetry.
 
  • #3
You don't say what D_1 and D_2 are and don't justify why it is that

dD_1/dD_2 = D_1/D_2

which would state that

log(D_1) = log(D_2) + K, ie D_1=kD_2

whatever D_i might be obviously.
 
  • #4
So what?
D_1 = k * D_2
and
k = F_2 / F_1 = M_2 / M_1 = const

D_i is a distance from the center of the lever.
 
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FAQ: Understanding Inert Bodies: Newton and Archimedes' Equations

1. What is an inert body?

An inert body is an object or substance that does not have any internal forces acting upon it and thus remains at rest or in a state of uniform motion.

2. Who were Newton and Archimedes?

Isaac Newton was a renowned English physicist and mathematician who is widely considered one of the most influential scientists in history. Archimedes was an ancient Greek mathematician, physicist, and engineer who is known for his contributions to the understanding of buoyancy and the principle of leverage.

3. What are Newton's equations for inert bodies?

Newton's first law of motion states that an object will remain at rest or in motion at a constant velocity unless acted upon by an external force. His second law of motion states that the net force acting on an object is equal to the product of its mass and acceleration. His third law of motion states that for every action, there is an equal and opposite reaction.

4. What is Archimedes' principle?

Archimedes' principle states that the buoyant force acting on an object submerged in a fluid is equal to the weight of the fluid that the object displaces. This principle is the basis for understanding the behavior of objects in fluids, such as boats floating in water.

5. How do Newton's and Archimedes' equations help us understand inert bodies?

Newton's equations provide a mathematical framework for understanding the behavior of objects in motion, including inert bodies. Archimedes' principle helps to explain the buoyancy of objects in fluids, which is an important concept in understanding the behavior of inert bodies in water or other fluids. Together, these equations help us to understand and predict the motion of inert bodies in various situations.

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