page_title: Mass
section_title: Inertial mass
hierarchical_section_title: Mass / Newtonian mass / Inertial mass
caption_attribution_description: Русский: Прибор для измерения инертной массы тела человека (массметр) в условиях невесомости (Государственный музей истории космонавтики имени К. Э. Циолковского)
context_page_description: Mass is both a property of a physical body and a measure of its resistance to acceleration when a net force is applied. An object's mass also determines the strength of its gravitational attraction to other bodies.
  The basic SI unit of mass is the kilogram. In physics, mass is not the same as weight, even though mass is often determined by measuring the object's weight using a spring scale, rather than balance scale comparing it directly with known masses. An object on the Moon would weigh less than it does on Earth because of the lower gravity, but it would still have the same mass. This is because weight is a force, while mass is the property that determines the strength of this force.
context_section_description: Inertial mass is the mass of an object measured by its resistance to acceleration. This definition has been championed by Ernst Mach and has since been developed into the notion of operationalism by Percy W. Bridgman. The simple classical mechanics definition of mass is slightly different than the definition in the theory of special relativity, but the essential meaning is the same.
  In classical mechanics, according to Newton's second law, we say that a body has a mass m if, at any instant of time, it obeys the equation of motion
  where F is the resultant force acting on the body and a is the acceleration of the body's centre of mass. For the moment, we will put aside the question of what "force acting on the body" actually means.
  This equation illustrates how mass relates to the inertia of a body. Consider two objects with different masses. If we apply an identical force to each, the object with a bigger mass will experience a smaller acceleration, and the object with a smaller mass will experience a bigger acceleration. We might say that the larger mass exerts a greater "resistance" to changing its state of motion in response to the force.
  However, this notion of applying "identical" forces to different objects brings us back to the fact that we have not really defined what a force is. We can sidestep this difficulty with the help of Newton's third law, which states that if one object exerts a force on a second object, it will experience an equal and opposite force. To be precise, suppose we have two objects of constant inertial masses m₁ and m₂. We isolate the two objects from all other physical influences, so that the only forces present are the force exerted on m₁ by m₂, which we denote F₁₂, and the force exerted on m₂ by m₁, which we denote F₂₁. Newton's second law states that
  where a₁ and a₂ are the accelerations of m₁ and m₂, respectively. Suppose that these accelerations are non-zero, so that the forces between the two objects are non-zero. This occurs, for example, if the two objects are in the process of colliding with one another. Newton's third law then states that
  and thus
  If |a₁| is non-zero, the fraction is well-defined, which allows us to measure the inertial mass of m₁. In this case, m₂ is our "reference" object, and we can define its mass m as (say) 1 kilogram. Then we can measure the mass of any other object in the universe by colliding it with the reference object and measuring the accelerations.
  Additionally, mass relates a body's momentum p to its linear velocity v:
  ,
  and the body's kinetic energy K to its velocity:
  .
  The primary difficulty with Mach's definition of mass is that it fails to take into account the potential energy (or binding energy) needed to bring two masses sufficiently close to one another to perform the measurement of mass. This is most vividly demonstrated by comparing the mass of the proton in the nucleus of deuterium, to the mass of the proton in free space (which is greater by about 0.239%—this is due to the binding energy of deuterium.). Thus, for example, if the reference weight m₂ is taken to be the mass of the neutron in free space, and the relative accelerations for the proton and neutron in deuterium are computed, then the above formula over-estimates the mass m₁ (by 0.239%) for the proton in deuterium.  At best, Mach's formula can only be used to obtain ratios of masses, that is, as m₁ /m₂ = |a₂| / |a₁|.  An additional difficulty was pointed out by Henri Poincaré, which is that the measurement of instantaneous acceleration is impossible: unlike the measurement of time or distance, there is no way to measure acceleration with a single measurement; one must make multiple measurements (of position, time, etc.) and perform a computation to obtain the acceleration.  Poincaré termed this to be an "insurmountable flaw" in the Mach definition of mass.
caption: Massmeter, a device for measuring the inertial mass of an astronaut in weightlessness. The mass is calculated via the oscillation period for a spring with the astronaut attached (Tsiolkovsky State Museum of the History of Cosmonautics)
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