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About force, mass, newton's 2nd law and inertial frame

  1. Dec 2, 2007 #1
    I was reading a physics textbook. It is stated that force is defined by mass and acceleration (a force of 1N causes a mass of 1kg to accelerate at 1ms^-2). In later part, it is stated that mass is defined by force and acceleration (a mass acted by a force of 1N accelerates at 1ms^-2 is 1kg). As force is defined by mass, mass has to be defined first. But mass is defined by force. So it's sth circular and I am confused.

    Also, Newton's 2nd law (and hence Newton's 1st law too) is only true for inertial frames. However, whether a frame is inertial is determined by whether Newton's 2nd law is true for that frame. Again it's sth circular and I'm am confused.

    Can anyone please clear my concept?
  2. jcsd
  3. Dec 2, 2007 #2

    D H

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    In the SI system, mass is defined first. The kilogram is a fundamental unit in the SI system. In fact, it is the only remaining fundamental unit that is still based on a prototype. The Newton (the SI unit of force) is a derived unit based on mass, distance, and time.

    The modern view is that Newton's first law defines the concept of inertial frames and introduces the concept of force. This law dictates the behavior (i.e., constant velocity) of an object that is not subject to any forces. This law does not define what a force is. It only posits behavior when no forces are present. So long as a body does move with a constant velocity when no forces are present, Newton's first law remains valid regardless of whether a force makes an object jump up and down or squirt sideways to avoid the force. Note well: A body with no net external forces does not move with a constant velocity in a non-inertial frame. Behavior given no net forces suffices to determine whether or not a reference frame is inertial.

    What Newton's second law does is specify behavior in an inertial frame (esablished by the first law) when the net forces acting on a body are other than zero. Note well that Newton's second law definition of force is very open-ended. It defines forces only in terms of what they do.

    Bottom line: No circularity. In SI, mass is fundamental, force is not. Note well: We could just as easily have made force a fundamental unit. That, along with distance and time, would require mass to be a derived unit. Newton's first law defines an inertial frame. Newton's second law defines force.
  4. Dec 2, 2007 #3


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    force = (change in mass * change in velocity) / change in time.

    That's why you need a reference point to build up these units, i.e. speed of light. So you'd start from the humble meter and work upwards.
  5. Dec 2, 2007 #4
    I'm not sure I agree entirely with DH. It's true that the standard used to define SI units of measurement is the kg and that the force of 1 N is derived from that. As he points out, though, it could have been the other way around, and I'd suggest that the main reason the kg is used as the standard is that it's much easier to create and maintain a 1 kg artifact than to create and maintain a 1 N artifact (how do you store a "force" in a box somewhere?).

    That all has to do with defining standards for measurement, however - it's all secondary to the fundamental definition of what the physical quantity is in the first place, which is what I believe lwymarie was asking about in the original post. Physical quantities are generally defined in terms of measurements, and we have very basic notions of how to measure length and time, but not of how to measure mass. We generally measure it by balancing forces, such as the force of gravity and the force of a spring in a scale, or by measuring the acceleration produced by a known force, which can be measured without first knowing mass, e.g. with a stretched calibrated spring.

    In school I learned that the second definition lwymarie gave is correct for mass, i.e. that mass is the ratio of force to acceleration, both of which may be defined and measured independently. The first definition is correct for defining a unit of measurement for forces, but not for defining what a force is, in my opinion.

    [By the way, I'm really talking about "inertial mass" here. "Gravitational mass" comes up in Newton's Law of gravity, but for the sake of measuring mass with a spring scale here, we can just take it as given that the weight of an object, i.e. the force of gravity on it, is proportional to its mass and not worry about the constant of proportionality.]
    Last edited: Dec 2, 2007
  6. Dec 2, 2007 #5


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    It is a good question. But as far as I know there is no good answer. Many years ago I read a (famous) book called "The Science of Mechanics" written by Ernst Mach (also famous) in the late 19th century. In the the book Mach is essentially trying to establish the "foundations" of newtonian mechanics and among other things give an answer to your question. However, as far as I understood when I read it (which, again, was quite a while ago) he is forced to conclude that there is no good answer, force and mass are in newtonian mechanics essentially defined in terms of each other giving rise to a "circular argument" as you have correctly observed.
    Also, as far as I know Newton was also aware of this problem so it is by no means a new question.
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