Mass

The mass (or inertial mass) represents the amount of matter that constitutes a material body, attributing dynamic characteristics (inertia) when the bodies are subject to the influence of external forces.

The mass does not correspond with the amount of substance, the physical quantity for which it has been introduced in the SI a fundamental quantity, the mole (symbol mol). The mass of a body is commonly determined by measuring its inertia which is opposed to a change in its state of motion or the gravitational attraction to other bodies by comparison with a sample (see balance).

Kilogram (unit of mass)

The international prototype of the kilogram, an artifact made of platinum-iridium, is kept at the BIPM under the conditions specified by the 1st CGPM in 1889 when it sanctioned the prototype and declared: this prototype shall henceforth be considered to be the unit of mass.

The 3rd CGPM (1901), in a declaration intended to end the ambiguity in popular usage concerning the use of the word “weight,” confirmed that:

The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram.

The word “weight” denotes a quantity of the same nature as a “force”: the weight of a body is the product of its mass and the acceleration due to gravity; in particular, the standard weight of a body is the product of its mass and the standard acceleration due to gravity.

The value adopted in the International Service of Weights and Measures for the standard acceleration due to gravity is 980.665 cm/s2, value already stated in the laws of some countries.

It follows that the mass of the international prototype of the kilogram is always 1 kilogram exactly. However, due to the inevitable accumulation of contaminants on surfaces, the international prototype is subject to reversible surface contamination that approaches 1 μg per year in mass. For this reason, the CIPM declared that, pending further research, the reference mass of the international prototype is that immediately after cleaning and washing by a specified method. The reference mass thus defined is used to calibrate national standards of a platinum-iridium alloy (Metrologia, 1994).

Center of mass

The center of mass of an object is the point at which the object can be balanced. Mathematically, it is the point at which the torques from the mass elements of an object sum to zero. The center of mass is useful because problems can often be simplified by treating a collection of masses as one mass at their common center of mass. The weight of the object then acts through this point.

The position of the center of mass of a collection of masses is given by:

Sum of clockwise moments about the center of mass = Sum of anti-clockwise moments about the center of mass

or, if all the masses were instead placed at the center of mass then they must give the same resultant turning moment about any point in the system as all of the individual moments added together.

\[\vec{r}_{cm}=\dfrac{\sum_i m_i\vec{r}_i}{\sum_i m_i}\]

How to find the center of mass

The center of mass of an object with a uniform density can often be found without calculation, but by instead just looking at the symmetry of the object. For a rod of uniform density, it is intuitive that the center of mass will be halfway along its length. We are equating the mass of the pieces either side of the point of the balance (in this case the tip of the wedge). This is the point at which the weight of the rod acts.

The most fundamental symmetry transformation for determining the position of the center of mass is rotational symmetry. If a massive object or collection of objects has rotational symmetry about a single point then that point will be the center of mass of the collection of objects.

Diagram showing the lines of reflectional symmetry (mirror lines – in grey) on a star.

An object or collection of objects with more than one line of reflectional symmetry will have its center of mass at the intersection of those lines. If the object only has one line of symmetry then the center of mass will be at some point along that line. If an object has a line of reflectional symmetry, then a mirror can be placed along that line and the “missing” half of the image will appear in the mirror.

Reflectional symmetry about a line or plane is less fundamental than rotational. If there are at least 2 lines of symmetry to form a point (which is the center of mass), then rotational symmetry about that point will also exist. Lines of reflectional symmetry are often more useful to think about than the point of rotation when determining the center of mass, as it is often a distance along with one of these lines that are used to describe the position of the center of mass. All the star’s lines of symmetry go through one point in the center, so this is the center of mass.

Negative mass

Negative mass is a hypothetical counterpart to ordinary (positive) mass. Such matter would violate one or more energy conditions and show some strange properties, stemming from the ambiguity as to whether attraction should refer to force or the oppositely oriented acceleration for negative mass.

Although it is not known if negative mass exists, or even if its existence is theoretically possible, several scientists have speculated on its properties. Among these are Hermann Bondi in the 1950s, Banesh Hoffman (1906-1986), of the City University, New York, in the 1960s and ’70s, and Robert Forward, in the context of spacecraft propulsion, in the 1980s. In both Newton’s and Einstein’s theories of gravity, negative mass is a requirement for antigravity to exist.

The concept of negative mass arises in the first instance by analogy with electric charges, of which there are both positive and negative varieties. Just as a positive electric charge can be canceled by a negative charge, thus giving rise to the possibility of screening against electric forces, so we can envisage the possibility of “gravity screens,” if negative mass existed to neutralize ordinary, positive mass.

References

  1. Bondi, H. “”Negative Mass in General Relativity,” Reviews of Modern Physics, Vol. 29, No.3, July 1957, pp. 423-428.
  2. Forward, R. L. “Negative Matter Propulsion”, Journal of Propulsion and Power (AIAA), Vol. 6, No. 1, Jan.-Feb. 1990, pp. 28-37.
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