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What Is The Formula For Mass?

Alex Bolano
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December 17, 2018
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Mass is a property of physical objects and a measure of how resistant that body is to acceleration. One can consider the mass of an object as a measure of how much physical “stuff” makes up that object.

Unlike relational properties such as position, velocity, or potential energy, which always must be defined with respect to another object or a reference point, mass is an intrinsic property that an object has independently of its relation to other things. The mass of an object can be calculated in a number of different ways:

All three of these formulae are a way of determining the mass of an object. Since mass is a fundamental property, It is not defined in terms of other units, like the joule (J) of newton (N) are. There are other ways to calculate the mass of an object, but these three formulae are the most common ones.

m=ρV

m=W/g

m=F/a

Units Of Mass

The SI-accepted unit for mass the kilogram (Kg). The kilogram is the only base SI unit with a prefix in its name (kilo-). Originally, one kilogram was defined as the mass of one cubic deciliter (dL) of water at its melting point. As of 1889, the kilogram was redefined as the mass of the International Kilogram Prototype (IPK), a physical artifact meant to be the universal reference mass for the kilogram. Originally the IPK was a weight made out of cast iron. Currently, the accepted IPK is a 39 mm tall cylinder made of a special platinum alloy.

“Words, too, have genuine substance – mass and weight and specific gravity.” — Tim O’Brien

As of 2018, the kilogram is the only SI unit that has a physical object as its reference value. All other SI units have been redefined in terms of fundamental physical constants, such as the speed of light or the Planck constant. In November 2018, the General Conference of Weight and Measures (GCPM) voted to redefine the kilogram in terms of fundamental physical constants, a change that will take hold on May 20, 2019.

Ways To Calculate Mass

From Density And Volume

The density of an object, sometimes represented by the Greek letter “ρ,” is a measure of mass per unit volume. Essentially, density tells you how tightly packed the mass of an object is. The denser an object is, the more mass per unit of volume it has.

For example, water has density of 977 kg/m3 at standard temperatures and pressures. That is, one cubic meter of water has a mass of 977 kg. If we know the density and volume of a substance, we can also figure out the mass of that substance. Say we have a 0.7m3 sample of water. How much mass does that sample have?

Solving for mass gives us:

m=ρV

m=(0.7m3)(977kg/m3) = 683 kg

0.5 cubic meters of water at standard temperature and pressure would have a mass of 683 kg.

Some objects are incredibly dense. A neutron star, for example, has an average density of 1.1 x 1018 kg/m3.  A single teaspoon of a neutron star would weigh about 100 million tons on Earth.

“Mass becomes immobile; it cannot manoeuvre and therefore cannot win victories, it can only crush by sheer weight.” — Hans Von Seeckt

From Force And Acceleration

The property of mass is also understood as the measure of a physical object’s resistance to being accelerated by the application of an external force. This concept of mass is sometimes referred to as inertial mass. Inertia is the tendency for moving body to continue in a constant state of motion, so inertial mass is a measure of how much inertia a body has and how difficult it is to change its state of motion. The relationship between mass, force, and acceleration is expressed by Newton’s second law of motion F=ma. This mathematical relationship tells us that, in the face of a constant force, a more massive body will accelerate more slowly. By measuring the force applied to a body and measuring the observed acceleration, we can calculate the mass of the body.

For example, say that we apply a 748 N force to a metal cube, and we measure its acceleration as 21m/s2. What is the mass of the metal cube? We can figure calculate the mass by dividing the magnitude of the force by the magnitude of acceleration so:

m=F/a

m=(748N)/(21m/s2) ≈ 35.62 kg

So we know that the metal cube must have a mass of 35.62 kg.

From Weight

Strictly speaking, weight and mass are two different things. In English, the two words “weight” and “mass” are synonymous, yet they have distinct meanings in the physical sciences. Mass is an invariant property that does not change from location to location. Weight is a measure of the gravitational field strength acting on a massive body. Since gravitational field strength can differ, i.e., the Moon has a weaker gravitational field strength than the Earth, an object’s weight can differ in different environments.

The relationship between mass and weight is given by W = mg, where g is the measure of acceleration due to gravity. The exact value of g differs depending on the location. On Earth, g has a value of approximately 9.81 m/s2 while on the moon, g is about 1.6 m/s2. The expression W=mg gives weight in terms of Newtons, while everyday understandings of weight are given in pounds (lbs) the conversion rate from newtons to pounds is about 1 N=0.22 lb.

For example, on the surface of the Earth where g=9.81 m/s2, a 50 kg object would have a weight in pounds of:

W=(50kg)(9.81m/s2)=490.5N

Converting newtons to pounds gives us:

490.5N(.22lbs/1N)≈108 pounds

Conversely, on the moon where g has a value of 1.6m/s2, a 50 kg object would weigh:

W=50(kg)(1.6m/s2)(0.22lbs/1N)≈18 pounds

The same 50 kg object weights 108 lbs on Earth and 18 lbs on the moon.

Likewise, if we know the weight of an object, we can work backward to figure out its mass. Say an object weighs 160 pounds of Earth. we can calculate the mass of the object as:

180lbs(1N/0.22lbs)=818.18N

818.18N=m(9.81m/s2)

m=818.18N/(9.81m/s2)≈83.4 kg

So a 180 lb body on Earth has a mass of about 84.3 kg.

Mass-Energy Equivalence

For a very long time, scientists believed that an object’s mass could be considered completely independent of its other properties. However, in the early 20th century, Einstein’s theory of special relativity showed that mass and energy are actually two different names for the same physical quantity. Specifically, the mass of an object and its total energy are related by Einstein’s famous equation E=mc2, where c is the speed of light in a vacuum.

E=mc2 tells us that the total energy of a non-moving body is directly proportional to its mass by a factor of c2. Since c=3,000,000 m/s, c2 is an extremely large factor. Consequently, even a tiny bit of mass contains an overwhelming amount of intrinsic energy. To put the matter in perspective, the total amount of energy from completely converting 1 gram of matter into energy is about equal to 21.5 kilotons of TNT—the strength of the Hiroshima atomic bomb.

“Any fool can know. The point is to understand.” — Albert Einstein

In certain physical processes, like nuclear fission or the movement of a body in an intense gravitational field, matter is converted to energy and released in the form of large quantities of light and heat.  Specifically, Einstein’s equation tells us how we can calculate the amount of energy released during such reactions.

Say 30 kg of uranium (Ur) is put into a nuclear reactor. During fission, approximately 0.1% of this mass is completely converted into energy. How much energy is yielded?

0.1% of 30 kg is 0.3 kg. Plugging this into Einstein’s equation gives us:

E=(0.3 kg)(3,000,000)2=(0.3)(8.98755179 × 1016)=2.69626554 × 1016 J

Completely converting just 0.3 kg (0.6 lbs) of matter into energy releases 2.69626554 × 1016 J of energy. This amount is about the same as detonating over 6 million tons of TNT (12 billion pounds), enough to completely level even the largest cities.

We can also work backward from some energy amount to determine the amount of mass that was converted. Say some fission reaction releases 1.6178 × 1016 joules of energy. How much mass was converted into energy during this process? Using our handy-dandy mass-energy equivalence formula we can determine:

1.6178 × 1016 J =m(8.98755179 × 1016)

(1.6178 × 1016)/8.98755179 × 1016)=m

m ≈ 0.18 kg

So about 0.18 kg of mass was converted into energy.

Why Do Objects Have Mass?

It has only been recently that scientists have begun to unlock the answer to why particles have mass at all. In the 1960s, several scientists noticed some problems with their equations describing the behavior of fundamental particles. Specifically, their equations predicted that certain particles generated during high-speed collisions would not have any mass. However, experimental observation indicated that these particles did, in fact, have a non-zero mass.

Scientists theorized that a boson’s mass could be generated by the interaction between those bosons and an all-pervading field called the Higgs field (in honor of one of its theorizers Peter Higgs). When massless bosons move against this field, their momentum is slowed, and they lose some energy. The Higgs field converts this energy into mass-energy, which manifests as the property mass that we measure. This interaction between the bosons and the Higgs field was predicted to create a new particle, a tiny boson dubbed the Higgs boson. The particle accelerator at CERN finally demonstrated the existence of the Higgs particle in 2013 and on October 8, 2013, Peter Higgs and François Englert were awarded the Nobel Prize in physics for their theoretical work on the particle.

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