Mass and Weight

FIELDS OF STUDY: Classical Mechanics; Geophysics

ABSTRACT: This article discusses the relationship between mass and weight. Mass is an intrinsic property of matter and exists independent of any outside influence. Weight is an extrinsic property of matter and depends on the interaction of mass with the external force of gravitational acceleration.

Principal Terms

• density: a measure of the mass of a quantity of matter relative to the volume of space that it occupies.
• extrinsic value: any characteristic property of matter that is due to the interaction of the matter with an external factor.
• gravitational acceleration: the rate of increase of velocity experienced by matter under the influence of a gravitational field; often referred to as the force of gravity.
• intrinsic value: any characteristic property of matter that is due solely to the nature of the matter itself.
• kilogram: the base unit of mass in the International System of Units (SI), equal to 1,000 grams (2.2 pounds).
• matter: anything that can be characterized as having mass and can be measured by some criterion of measurement.
• pound: a standard unit of mass in the imperial and US customary measurement systems, equivalent to 0.45359237 kilograms; also used as a unit of force.
• slug: a unit of mass in the foot-pound-second (FPS) system, defined as the amount of mass that will experience an acceleration of one foot per second squared under one pound of force; equivalent to approximately 14.5939 kilograms or 32.1740 pounds.

Intrinsic and Extrinsic Properties

Physics is the study of the interaction of matter with its environment. Matter is defined as anything that has mass and can be measured in some way. The base unit of matter is the atom. An atom consists of the three principal subatomic particles, called protons, neutrons, and electrons. All of the matter in the known universe is made up of a limited number of different atoms. The number of protons in each atom determines its identity as a chemical element. While as many as 118 different elements have been identified, only 98 are known to occur naturally, albeit some only in trace amounts and for infinitesimal periods of time. Of these elements, the simplest two, hydrogen and helium, are the most common. Hydrogen and helium make up more than 99 percent of all observable matter in the universe.

How different atoms interact determines the unique properties of any type of matter. Some properties are common to all forms of matter, due to the simple fact of its physical existence. Such properties are called intrinsic properties. Intrinsic properties are constant regardless of any external factors. Mass is one such property; all atoms have mass. The mass of any amount of matter is the sum total of the mass of the atoms that make up that matter. Density is another property common to all matter. Density is determined by both the mass of the matter and the volume of space that it occupies.

Some properties of matter are affected by the influence of external forces. These properties are known as extrinsic properties. Weight is one example of an extrinsic property. Matter has weight because its mass is affected by gravitational acceleration, or the force of gravity. Though the mass of any matter is universally constant, its weight varies depending on the amount of gravitational acceleration it experiences. The most common form, or isotope, of hydrogen consists of a single proton, and so its mass is equal to the mass of one proton everywhere in the universe. One mole (approximately 6.022×10²³ atoms) of this isotope, which is known as hydrogen-1 or protium, has a mass of exactly 1 gram. On Earth, the weight of one mole of protium is also 1 gram. On the moon or on Jupiter or anywhere else in the universe, one mole of protium will have the exact same mass, but its weight will be different due to differences in gravitational acceleration.

The relationship between mass and weight is defined by the equation W = mg, where W is weight, m is mass, and g is the standard acceleration due to gravity. On Earth, the mass of a quantity of matter is generally the same as its weight, because one unit of standard gravity (g) has been defined as equal to the acceleration due to Earth’s gravity, which is approximately 9.81 meters per second per second (m/s²). Therefore, to use this equation to calculate weight elsewhere in the universe, the gravitational acceleration in that location would have to be converted to standard gravity units, according to the equation 1 g = 9.81 m/s².

Units of Mass and Weight

In the British or imperial system of measurement, which is the system most commonly used in the United States, the base unit of length is the foot, and the base unit of mass and weight is the pound. In the International System of Units (SI), the modernized form of the metric system, the base unit of length is the meter, and the base unit of mass and weight is the kilogram. The foot-pound-second (FPS) system is a variation on the imperial system that was historically used for scientific applications prior to the adoption of the metric or SI system.

In the FPS system, the pound can serve as a unit of either mass or force. To distinguish one from the other, these units are referred to as pound mass and pound force. The pound mass is the same as the pound used to measure mass in the imperial system and is equal to 0.45359237 kilograms. The pound force is defined as the weight of one pound mass experiencing the acceleration of one unit of standard gravity, following the equation W = mg. The unit of mass that corresponds to pound force is the slug. The slug is defined as the amount of matter that will accelerate at one foot per second per second (ft/s²) when propelled by one pound force. As the SI system became more widely accepted, most of these nonstandard units became obsolete.

Weights and Masses in Other Places

Gravity is one of the four fundamental forces in nature. The other three fundamental forces are the electromagnetic force, the weak nuclear force, and the strong nuclear force. Gravity is an intrinsic property of matter. Every atom exerts gravitational force. The magnitude of an atom’s gravitational force is determined solely by its mass. However, gravity is the weakest of the four forces. A great deal of mass is required before the effects of gravity can be felt. The gravitational force exerted by a planet is the cumulative sum of the gravity of every atom in that planet’s structure. The more mass a planet has, the greater its gravity and the greater the weight of any mass in its gravitational field.

Yet the total gravitational force of a planet does not necessarily correspond to the amount of gravity experienced on that planet’s surface. Gravity is a function of both mass and distance, where distance is measured from the center of the object exerting the force. Thus, the mass of Saturn is more than ninety-five times that of Earth, but its surface gravity is only slightly greater than on Earth, approximately 11.44 m/s². This is because the distance from Saturn’s surface to the center of its mass is so great.

Sample Problem

A celestial explorer on Mars collects a measured weight of 50 pounds of Martian rock samples and brings them back to Earth. The gravitational acceleration of Earth is approximately 9.81 m/s², but that of Mars is only about 3.69 m/s². What is the measured weight of the Martian rocks on Earth in grams? Recall that 1 pound equals 0.45359237 kilograms, or 453.59237 grams.

Answer:

The weight of an object is the product of its mass and the gravitational acceleration that it experiences. Since the mass of the rocks is the same on Earth as on Mars, their weight will depend on the gravitational acceleration that they experience. Their weight on Mars is given by the expression

W_M = ma_Mwhere W_M is the weight of the rocks on Mars, m is the mass of the rocks, and a_M is the gravitational acceleration on Mars. The same equation can be used to determine the weight of the rocks on Earth, substituting W_E for W_M and a_E for a_M:

W_E = ma_E

Because the mass, m, is constant, the two equations can each be solved for m and then combined into one:

Solve the equation for W_E:

To convert this weight to grams, multiply the weight in pounds by the conversion factor of 453.59237 grams per pound:

W_E = (132.927 lb)(453.59237 g/lb)

W_E = 60,294.673 g

The Martian rocks weigh approximately 60,295 grams on Earth.

Mass, Momentum, and Energy

One of the dangers of space exploration is impact by micrometeoroids. Without proper shielding, a micrometeoroid the size of a grain of sand can blast a hand-sized hole in the outer wall of a shuttle. This damage is due to the kinetic energy of the micrometeoroid. Kinetic energy is a property that depends on both its mass and its velocity, according to the equation

where E_k is the kinetic energy, m is the mass, and v is the velocity at which the mass is traveling. Even though the mass of a micrometeoroid is very small, the relative velocity of such particles is typically around 10 kilometers per second, or 36,000 kilometers (22,369 miles) per hour, giving them a very large amount of kinetic energy.

Another intrinsic property that is closely related to kinetic energy is momentum. An object’s momentum is also defined by its mass and its velocity, according to the expression

p = mv

where p is momentum, m is mass, and v is velocity. In mathematical terms, the momentum of an object is said to be the derivative of its kinetic energy with respect to velocity. The law of conservation of energy states that when two or more objects collide, any energy lost by one object must be gained by the other object(s). That transfer of energy will be reflected in their relative speeds before and after the collision.

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