View Full Version : How is F=ma derived
2005-Sep-08, 07:50 AM
F=ma is used frequently in physics, and is dimensionally correct. But how is it derived?
2005-Sep-08, 09:31 AM
At the most elemetary level, from obsevation. At the advanced level.. I'll tell you when I get there... ;)
OK, I share Matt's query... how is it derived?
2005-Sep-08, 11:11 AM
In Newtonian Mechanics, F = ma -- or better, F = dp/dt (where p is the vector momentum) -- is a postulate inferred from observations, and the force F has an empirical definition.
The acceleration a is well defined within calculus, while the concept of force is based on an intuitive definition.
In this case, force is not strictly defined theoretically, and we know that the "intuitive" forces (for example, weight or springs) are vectorial quantities from experiments.
The lack of a more rigorous definition of force is a weak point of Newtonian Mechanics from a purely theoretical point of view.
In Lagrangian Mechanics, one can define mathematically a function linked to translations in time (a change of the origin for the time coordinate) using Noether's theorem.
This function turns out to be the energy, from which the force is derived as spatial gradient (usually the forces considered are conservative, so he function "energy" is well defined).
This way force is more rigorously defined theoretically.
In Hamiltonian Mechanics, the Hamiltonian function (which is the energy) determines the evolution in time, and we do not actually use Newton's second law explicitly.
2005-Sep-08, 12:02 PM
As Papageno said, force is the change in momentum ( p) with respect to time, or using the terminology of differential calculus, F = dp/dt. Momentum is defined as mass x velocity. Using the product rule, dp/dt = d(mv)/dt = m(dv/dt) + v(dm/dt). dv/dt is the definition of acceleration , F = ma + v(dm/dt). In situations where the mass doesn't change, dm/dt = 0, so the equation simplifies to F = ma. It is important to remember that this simplified formula only works when mass doesn't change, or the change is so small that the term v(dm/dt) is negligible. When you are dealing with a car for instance, the mass is changing because fuel is being burned and is expelled from the tailpipe, but this change is miniscule enough to ignore. However, if you are doing calculations involving a rocket, the change in mass is rather substantial because the fuel is burned so quickly, and must be accounted for. Given that the space shuttle loses several million kilograms in a few minutes, the v(dm/dt) term needs to be used. Also, if you are dealing with relativistic velocities, where mass increases with velocity, the v(dm/dt) term becomes very important.
2005-Sep-09, 04:17 PM
As far as I know m (mass) was defined by Newton as a property of a particle, and F (force) as a property of a directed pair of particles. Newton postulated that:
The acceleration of any particle can be found by dividing the total force on that particle by the mass of that particle.
The total force on any particle is the sum of the forces of all the directed pairs where the particle under examination is the first member of the pair.
The force of any directed pair of particles is the negative of the force of the inverted pair.
A (acceleration) is of course the rate of change in v (velocity) and v is the rate of change in s (location).
Some letters (like F) should be bold, but I do no know how to embolden letters in my text. I could do it before the two fora merged, but now I can't discover how to do it.
2005-Sep-10, 11:03 PM
I guess we have all heard different stories how the formula was derived. Here's the rumor that I heard..
Newton took Kepler's elliptical orbit...He placed small tangents to that ellipse and gave each a vector arrow with a given length that described where the planet would go if no other influence was upon it...Inertia... At the point each tangent shares with the ellipse he draws another vector aimed right at the sun, shorter than each tangeant vector (each 'inertia' vector) since the influence of the sun is weaker than the influence of inertia...
The influence of the sun changes the motion of the planet away from its inertial intent so he labeled that vector "a"...He labeled the tangent vector "m" and the resulting parallelogram provided two things: The diagonal was the resulting path and the area of the parallelogram was the area of responsibility...."F"...The area of the parallelogram can be figured out by setting F=ma..
The smaller that "m" is to "a", the less effort to accelerate an object and the more deviation can take place in the inertial path...
He needed to make sure that no forces within the planet influenced its own path..So he made an experiment with 3 partitions that would float in water..On one partition he strapped a magnet...On another he strapped an iron bar..The third partition was placed between the magnet's partition and the iron's partition in the water..If the force exerted by the magnet were stronger, the whole assembly would be pulled toward the magnet and the float would move through the water always following the magnet side of the apparatus..If the force exerted by the iron were the stronger, the whole assembly would be pulled toward the iron..Since the system just stayed at rest in the pond and didn't move at all, he concluded that the force of each balanced out. Changing the strength of the magnet didn't change the results of the experiment..Therefore, whatever volacanic activity exists (or any internal forces) in any given planet has no influence on its journey and the parallelogram's diagonal does describe the path of the planet...Further, we have realized Newton's 3rd Law from this last experiment..
Galileo's ramp experiments gave him his first law..
Einstein realized all of Kepler's and Newton's laws from the geometry of world lines..
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