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Why did Newton define force as F=ma instead of stuff like F=m*(a^2) or F=(m^2)*a?
Newton did not simply "define" F=ma, he made observations and based his results on those observations.
Then what is the definition of force?
Then what is the definition of force?
Nature defined it that way, not Newton.Why did Newton define force as F=ma instead of stuff like F=m*(a^2) or F=(m^2)*a?
Nature defined it that way, not Newton.
Force is the thing that causes a body's motion to change.
http://dictionary.reference.com/browse/define" [Broken]: to explain or identify the nature or essential qualities of;Nature doesn't define anything.
The question was why Newton defined force as mass x acceleration (second law), not why he defined it as something that changes the motion of a body (first law).That's what Newton says, but according to Aristoteles force is the thing that causes a body's motion not to change. Because using the same word with different sense is not acceptable in science a clear definition was necessary. This definition is given by Newton's laws of motion.
http://dictionary.reference.com/browse/define": to explain or identify the nature or essential qualities of;
Do humans define the speed of light and make it constant for all inertial observers? Do humans define Galilean relativity? Or does nature do this and leave humans to just work out the units? Perhaps it is just a different way of looking at it.
The question was why Newton defined force as mass x acceleration (second law), not why he defined it as something that changes the motion of a body (first law).
Newton's first law implies that "force" is something that changes the motion of a body: in the absence of force, there is no change in a body's motion. So a body at rest in one inertial frame of reference moves at constant speed relative to another inertial frame of reference.
To derive the second law one need only accept:
#1: that two forces are equal if they effect equal changes in the motion of a body in equal times; and
#2: all inertial frames of reference are equivalent (Galilean relativity)
Suppose you have a body, A, at rest and you subject it to force F for a time t giving it a speed v. The change in motion is v-0 = v. Now it is at rest relative to a frame of reference moving at velocity v relative to the initial frame of reference. I then consider the new frame of reference as the "rest" frame and I repeat the same experiment: i.e. I apply the same force F again for a time t. Again the change in motion (principle #1) must be the same, so the velocity again changes by v, but this time it is relative to the new inertial reference frame.
So by applying the force F for 2t I get a change in motion of 2v. I can repeat this experiment over and over and we can see that Δv/Δt = constant if the force is constant and mass is constant (ie. same body).
Then I add another identical body to the first (ie. I double the mass) and I apply the same force F to each body simultaneously, so I have a force of 2F. This is identical to the first experiment except that I have two bodies and two forces. So the change in motion in time t has to be the same as the first: v. Pretty soon we realize that force varies as mass and as Δv/Δt. And that is pretty much the second law.
Now was that the handiwork of nature or of Newton?
AM
Newton did not define force as F=ma. That is an after-the-fact refinement of Newton's laws that took about 200 years to develop.Why did Newton define force as F=ma instead of stuff like F=m*(a^2) or F=(m^2)*a?
If I physically add another identical body, I have doubled the quantity of matter. If I weigh the two together, they have double the weight of one alone. So I pick a standard body (eg. a gram). I compare the quantity of matter in an object to the quantity of matter in the standard body (using, say, a scale) and I do an experiment to show whether they behave the same way in response to the same force. I see that they do. So I call this property mass.But how do you define the mass, ie. how do you know you've doubled the mass?
Engineers.Who needs the idea of a force?
Yeah. I don't think I would want an office on the 10th floor of a building designed by an engineer who did all the calculations using the Hamiltonian formulation of mechanics.Engineers.
I challenge you to come up with a Lagrangian or Hamiltonian formulation for a rotating turbine blade that is just on the verge of cavitating, for modeling the lift and drag on a supersonic jet, or for modeling the behavior of a space vehicle with rockets that have thrust (thruster on/off, and when on, magnitude and direction) controlled by some automated control system.
What a really mean to say is that the idea of a force doesn't really provide any new physical insight. All the physics is contained in the Lagrangian. Of course, from a practical point of view, engineers need to know about forces. But this is a physics forum, not an engineering forum. From a physicist point of view, forces don't add much to our physical understanding of the world.
Baloney. There are places where you cannot construct a Lagrangian. The real world is messy, and physicists are called upon to help analyze problems in the real world. A physicist who is asked to consult with NASA had better be able to deal with the world of forces and torques because that is the world in which NASA works for the most part. A physicist who is asked to help formulate a high-precision model of the solar system had better be able to deal with forces (with general relativity modeled as perturbative forces) because a full-blown GR solution to the N-body problem is downright intractable.What a really mean to say is that the idea of a force doesn't really provide any new physical insight. All the physics is contained in the Lagrangian. Of course, from a practical point of view, engineers need to know about forces. But this is a physics forum, not an engineering forum. From a physicist point of view, forces don't add much to our physical understanding of the world.
http://dictionary.reference.com/browse/define" [Broken]: to explain or identify the nature or essential qualities of;
Do humans define the speed of light and make it constant for all inertial observers? Do humans define Galilean relativity? Or does nature do this and leave humans to just work out the units? Perhaps it is just a different way of looking at it.
That is incorrect. Quantum mechanics, for example, says the photon has zero intrinsic mass. In every experimental attempt to measure the mass of the photon, the measured value has been extremely small and has been consistent with no intrinsic mass. All that improving experimental techniques has done is to make the measured range of the photon's rest mass ever smaller -- and still consistent with a null intrinsic mass.I've often heard that all our theories are human inventions. They do not describe Nature, since all of them are provably wrong.
That is incorrect. Quantum mechanics, for example, says the photon has zero intrinsic mass. In every experimental attempt to measure the mass of the photon, the measured value has been extremely small and has been consistent with no intrinsic mass. All that improving experimental techniques has done is to make the measured range of the photon's rest mass ever smaller -- and still consistent with a null intrinsic mass.
You are perhaps confusing falsifiability with falsification. Any scientific theory should be falsifiable. A viable scientific theory must not have been falsified, at least not within its domain of applicability. Were that to happen, the experiment would be examined critically (e.g., OPERA results). If the results held, there are only three possible outcomes: Toss the theory out (example: caloric theory), reduce the domain of applicability of the theory (example: Newtonian mechanics), or modify the theory so that it accommodates the new results (example: neutrino oscillations and the standard model).
Newton already had a somewhat vague definition of force, and he ways to measure forces independent of acceleration.
Newton attributed his first two laws to Galileo. Galileo's experiments showed that force is proportional to (modern view) the product of mass and acceleration. Newton did even more experiments in this regard.
Sorry if this is considered necroposting! :uhh:
I just wanted to see if you could direct me to resources that show how Newton measured force and of Galileo and Newton's experiments that showed force being proportional to the product of mass and acceleration.
When two falling unequal masses accelerate at the same rate, in the absense of air resistance, this is because the greater mass experiences a greater gravitational force being resisted by a greater inertia, and the smaller mass experiences a smaller gravitational force being resisted by a smaller inertia.
Their common and constant acceleration a = the ratio F1/m1 = the ratio F2/m2.
This is experimental confirmation that a=F/m, which is the same thing as F=ma.
If the claim was that "falling unequal masses accelerate at the same rate", you don't need to measure the force to prove it, you just need to simultaneously drop several objects of different weight/mass and see if they hit the ground at the same time. Which he is believed to have done, though the specifics of the experiments are in doubt: http://en.wikipedia.org/wiki/Galileo_Galilei#Falling_bodies
http://en.wikipedia.org/wiki/Inertia#Classical_inertia
How far he took the logic I'm not sure, but if we know acceleration due to gravity is constant, then a=g=f/m
He figured out that friction was tripping-up Aristotle. Once you eliminate friction, it is easy to see that force causes acceleration and lack of force means no acceleration.If you don't need to measure the forces, how did Galileo know to attribute the accelerations to the forces?
Galileo attributed changes in motion to forces in a general way. But, at least in my understanding, it was not Galileo but Newton who attributed gravitational accelerations to gravitational forces. Galileo determined that in the absence of friction or resistance, all objects fall at the same rate. He determined that the relationship between time, t, of fall and height, h, of fall of an object was [itex]h = at^2/2[/itex]. But Galileo did not conclude that they fall at the same rate due to gravitational force being proportional to mass.But in order to make the proportion [itex]\frac{F_1}{m_1}\textit{=}\frac{F_2}{m_2}[/itex], as mikelepore stated, wouldn't force need to be quantified in some way? If you don't need to measure the forces, how did Galileo know to attribute the accelerations to the forces?
Galileo attributed changes in motion to forces in a general way. But, at least in my understanding, it was not Galileo but Newton who attributed gravitational accelerations to gravitational forces. Galileo determined that in the absence of friction or resistance, all objects fall at the same rate. He determined that the relationship between time, t, of fall and height, h, of fall of an object was [itex]h = at^2/2[/itex]. But Galileo did not conclude that they fall at the same rate due to gravitational force being proportional to mass.
AM
Why did Newton define force as F=ma instead of stuff like F=m*(a^2) or F=(m^2)*a?