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 Mass is Not Measured But Calculated

(For intro and history of Mass & Weight, see  Mass  and  Weight : There's a Difference)

08-15-23
 The Weight of any object is quite easy to obtain by using any assortment of available scales which are spring scales, counterbalance scales, or digital scales.  But what about Mass ?  If mass is inherent, and its value never changes, isn’t that more important to know ?  Although a bit confusing, it may be considered a bit of a paradox that weight is relative and changing and more significant in practical day-to-day living, whereas mass is an absolute entity but perhaps only significant to a discrete group of scientists. 

Although mass is most likely never needed in day-to-day circumstances, it may be absolutely required in certain science applications.  One example is chemistry where precise quantities are needed to obtain mixtures.  Since the value of mass is constant and considered an inherent property, obtaining mass instead of weight becomes of interest.  Other examples are in dynamical calculations to determine forces and energy of objects in motion such as trains, boats, airplanes, and cars; and space vehicles which travel through space and orbit the sun, earth, and moon.  Sometimes the term ‘inertial mass’ is used for motion studies to indicate the resistance an object has when being put in motion - which isn’t due to gravity -  but from its own  inherent “inertial” mass.  (The object is “sluggish”, thus the mass unit ‘slugs’).

.. we do not really know the mass of any celestial body in space, including all the asteroids, meteors, comets, stars, planets, and moons – not even our own Moon or Sun.  Not even our own Earth.

Mass is of interest to the scientist or engineer but is not ordinarily understood (nor needed) by the common layperson.  Mass is a bit more elusive than realized.  In fact, it may come as a shock that we don’t really know the mass of our own Earth, Moon, or Sun.  The values that are given are only estimates, that is, “ballpark figures”.  We are just now acquiring some idea how to determine masses of common asteroids due to the recent NASA DART mission.  It requires “bumping” the asteroid with an impacting spacecraft of known mass and velocity and analyzing its motion afterwards.  Otherwise, we do not really know the mass of any celestial body in space, including all the asteroids, meteors, comets, stars, planets, and moons – not even our own Moon or Sun.  Not even our own Earth.

To determine the mass of an object on earth one needs to know the value of g, the acceleration of gravity at the particular location.  The acceleration of gravity varies around the Earth and at different altitudes, which is why you can’t determine mass directly from a scale.  You need to know ‘g’ at your location.  This is why weigh scales should not really show mass units on a scale.  Mass will never register correctly when attempting to measure it directly from a scale.  It simply won’t be true nor correct.

Mass is obtained from the relation W = mg and solved as m = W/g.  The value of g can be obtained at a particular location by using scientific methods.  One of these is dropping a weight in a vacuum from a known distance and accurately timing its free fall.  From the Kinematics of Physics we have the relation:

x(t) = x0 + v0t + ½at2  

In this case, ‘g’ is in place of ‘a’ and since we are dropping a weight from rest at a known distance, v0 = 0 and x0 = 0, and x(t) is simply the known distance ‘d’. Therefore, simplifying the above

g = 2d/t2

   

A lab experiment to determine the acceleration of gravity g at its location.  A steel ball is dropped a known distance d from rest in an evacuated tube.  A light sensor is wired to a timer which is triggered when the ball drops.  This will accurately measure the time the steel ball free-falls within its gravity field from rest to the distance d at the light sensor.  The evacuated tube eliminates any error of air resistance.

 

After dropping a weight in a vacuum and measuring the time it takes to pass the light sensor from a known height, we can determine g.  We can now obtain the mass of any item or object by weighing the item at the same location and dividing by g, or  m = W/g.  The mass of the object will be the same anywhere on Earth (or anywhere in the Universe) but its weight will vary according to its location on Earth.  Both are valid entities, weight is relative and mass is absolute.  The weight of any item will always read correctly on a scale anywhere on Earth, just that it will vary according to g.  The mass, however, must be calculated and is not obtained directly from a scale.  However, once obtained it will remain the same anywhere on Earth and is a regarded a property of matter.

There exists other methods to obtain the acceleration of gravity g, but also require lab conditions.  A second method is the swinging pendulum, such as a weight on a string attached to a frictionless pivot, the entire apparatus also contained within a vacuum.  The period T is measured and incorporated into the formula for the swinging pendulum  T = 2π√L/g,  and solved as g = 4π2L/T2.  This is another method to find g, but along with an evacuated containment, a near frictionless pivot is needed to suspend a near-massless string.

Another, or third method, is by collision theory of Mechanics which is based on the Conservation of Momentum.  A familiar example is the motion of billiard balls when they hit each other and deflect according to exact laws.  (Lab conditions can perhaps improve on providing a more-or-less frictionless billiards table).   Determining mass by collision is what recently took place in the NASA DART mission of 2022, designed with the purpose of deflecting a possible hazardous asteroid.  To successfully deflect an asteroid, you must know the asteroid’s mass to a fair-degree-of-accuracy and then decide how much impact is needed to deter it.  The principles of the Conservation of Momentum are applied and the mass and speed of the impactor spacecraft are tailored to the specific asteroid.

Some might contend that g can be obtained from Newton’s universal law of gravity, or

F = G(m1m2)/r2

where G = 6.674 x 10-11 m3kg-1s-2  is the universal gravity constant, and m1 & m2 are the two attracting masses, one of which is the mass of the Earth and the other the falling body, and r the distance between the two masses.  Substituting W = mg for the force F (weight is a force) solves as:

g = GM/r2

where M is now the mass of the Earth (the smaller mass m algebraically cancels out).  Although appearing as a nice concise formula for determining g, there is yet one drawback which was already mentioned: We don’t really know the mass of the Earth (it is given as 6 billion trillion tons).1

 

1. Other estimates of the Earth's mass are derived from an average density, wherein it is arbitrarily carried out to several digits such as 5515 kg/m3.  Since the volume of the Earth is not so difficult to determine, it appears that the mass of the Earth is then adequately calculated.  However, a figure such as 5.9722 x 10^24 kg for the Earth's mass is regarded an academic undertaking.