Classical mechanics

Newton's Second Law

History of ... Fundamental concepts Space · Time · Mass · Force Energy · Momentum

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Mass is a fundamental concept in physics, roughly corresponding to the intuitive idea of how much matter there is in an object. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the Early Ideas on Motion The Greek philosophers, and Aristotle in particular were the first to propose that there are abstract principles governing nature Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another For other uses see Time (disambiguation Time is a component of a measuring system used to sequence events to compare the durations of In Physics, a force is whatever can cause an object with Mass to Accelerate. In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Matter is commonly defined as being anything that has mass and that takes up space. Mass is a central concept of classical mechanics and related subjects, and there are several definitions of mass within the framework of relativistic kinematics (see mass in special relativity and mass in General Relativity). Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects The term Mass in Special relativity usually refers to the Rest mass of the object which is the Newtonian mass as measured by an observer moving along with The concept of Mass in General relativity (GR is more complex than the concept of Mass in special relativity. In the theory of relativity, the quantity invariant mass, which in concept is close to the classical idea of mass, does not vary between single observers in different reference frames. See also Inertial frame A frame of reference in Physics, may refer to a Coordinate system or set of axes within which to

In everyday usage, mass is more commonly referred to as weight, but in physics and engineering, weight means the strength of the gravitational pull on the object; that is, how heavy it is, measured in units of force. In the Physical sciences weight is a Measurement of the gravitational Force acting on an object In Physics, a force is whatever can cause an object with Mass to Accelerate. In everyday situations, the weight of an object is proportional to its mass, which usually makes it unproblematic to use the same word for both concepts. However, the distinction between mass and weight becomes important for measurements with a precision better than a few percent (due to slight differences in the strength of the Earth's gravitational field at different places), and for places far from the surface of the Earth, such as in space or on other planets. In the physical sciences, Mass and Weight are different properties

Contents 1 Units of mass 2 Inertial and gravitational mass 2.1 Inertial mass 2.2 Gravitational mass 2.3 Equivalence of inertial and gravitational masses 3 See also 4 References 5 External links

Units of mass

In the SI system of units, mass is measured in kilograms, kg. Many other units of mass are also employed, such as:

• gram: 1 g = 0. For other uses of the words gram or gramme see Gram (disambiguation. 001 kg
• tonne: 1 tonne = 1000 kg
• atomic mass unit
• Planck mass
• solar mass
• eV/c²

Outside the SI system, a variety of different mass units are used, depending on context. This article is about the tonne or metric ton For other tons see Ton. The unified atomic mass unit ( u) or Dalton ( Da) or sometimes universal mass unit, is an unit of Mass used to express The Planck mass is the unit of Mass, denoted by m P in the system of Natural units known as Planck units. The solar mass is a standard way to express Mass in Astronomy, used to describe the masses of other Stars and galaxies.

Because of the relativistic connection between mass and energy (see mass in special relativity), it is possible to use any unit of energy as a unit of mass instead. The term Mass in Special relativity usually refers to the Rest mass of the object which is the Newtonian mass as measured by an observer moving along with For example, the eV energy unit is normally used as a unit of mass (roughly 1. 783 × 10^-36 kg) in particle physics. Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them A mass can sometimes also be expressed in terms of length. Here one identifies the mass of a particle with its inverse Compton wavelength (1 cm^-1 ≈ 3. 52×10^-41 kg).

For more information on the different units of mass, see Orders of magnitude (mass). To help compare different orders of magnitude, the following list describes various Mass levels between 10&minus36&thinsp kg and 1053&thinspkg

Inertial and gravitational mass

One may distinguish conceptually between three types of mass or properties called mass:^[1]

• Inertial mass is a measure of an object's resistance to changing its state of motion when a force is applied. In Physics, a force is whatever can cause an object with Mass to Accelerate. An object with small inertial mass changes its motion more readily, and an object with large inertial mass does so less readily.
• Passive gravitational mass is a measure of the strength of an object's interaction with a gravitational field. A gravitational field is a model used within Physics to explain how gravity exists in the universe Within the same gravitational field, an object with a smaller passive gravitational mass experiences a smaller force than an object with a larger passive gravitational mass.
• Active gravitational mass is a measure of the strength of the gravitational field due to a particular object. For example, the gravitational field that one experiences on the Moon is weaker than that of the Earth because the Moon has less active gravitational mass.

Although inertial mass, passive gravitational mass and active gravitational mass are conceptually distinct, no experiment has ever unambiguously demonstrated any difference between them. In classical mechanics, Newton's third law implies that active and passive gravitational mass must always be identical (or at least proportional), but the classical theory offers no compelling reason why the gravitational mass has to equal the inertial mass. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects That it does is merely an empirical fact.

Albert Einstein developed his general theory of relativity starting from the assumption that this correspondence between inertial and (passive) gravitational mass is not accidental: that no experiment will ever detect a difference between them (the weak version of the equivalence principle). Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 The equivalence principle However, in the resulting theory gravitation is not a force and thus not subject to Newton's third law, so "the equality of inertial and active gravitational mass [. . . ] remains as puzzling as ever". ^[2]

Inertial mass

This section uses mathematical equations involving differential calculus. Differential Calculus, a field in Mathematics, is the study of how functions change when their inputs change

Inertial mass is the mass of an object measured by its resistance to acceleration.

To understand what the inertial mass of a body is, one begins with classical mechanics and Newton's Laws of Motion. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the Later on, we will see how our classical definition of mass must be altered if we take into consideration the theory of special relativity, which is more accurate than classical mechanics. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial However, the implications of special relativity will not change the meaning of "mass" in any essential way.

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 force acting on the body and v is its velocity. In Physics, a force is whatever can cause an object with Mass to Accelerate. In Physics, velocity is defined as the rate of change of Position. For the moment, we will put aside the question of what "force acting on the body" actually means.

Now, suppose that the mass of the body in question is a constant. This assumption, known as the conservation of mass, rests on the ideas that (i) mass is a measure of the amount of matter contained in a body, and (ii) matter can never be created or destroyed, only split up or recombined. The law of conservation of mass/matter, also known as law of mass/matter conservation (or the Lomonosov - Lavoisier law says that the Mass of These are very reasonable assumptions for everyday objects, though, as we will see, mass can indeed be created or destroyed when we take special relativity into account. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial Another point to note is that, even in classical mechanics, it is sometimes useful to treat the mass of an object as changing with time. For example, the mass of a rocket decreases as the rocket fires. A rocket or rocket vehicle is a Missile, Aircraft or other Vehicle which obtains Thrust by the reaction of the However, this is an approximation, based on ignoring pieces of matter which enter or leave the system. In the case of the rocket, these pieces correspond to the ejected propellant; if we were to measure the total mass of the rocket and its propellant, we would find that it is conserved.

When the mass of a body is constant, Newton's second law becomes

where a denotes the acceleration of the body.

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 A and B, with constant inertial masses m_A and m_B. We isolate the two objects from all other physical influences, so that the only forces present are the force exerted on A by B, which we denote f_AB, and the force exerted on B by A, which we denote f_BA. As we have seen, Newton's second law states that

and

where a_A and a_B are the accelerations of A and B 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

Substituting this into the previous equations, we obtain

Note that our requirement that a_A be non-zero ensures that the fraction is well-defined.

This is, in principle, how we would measure the inertial mass of an object. We choose a "reference" object and define its mass m_B 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.

Gravitational mass

Gravitational mass is the mass of an object measured using the effect of a gravitational field on the object.

The concept of gravitational mass rests on Newton's law of gravitation. Newton 's law of universal Gravitation is a physical law describing the gravitational attraction between bodies with mass Let us suppose we have two objects A and B, separated by a distance |r_AB|. The law of gravitation states that if A and B have gravitational masses M_A and M_B respectively, then each object exerts a gravitational force on the other, of magnitude

where G is the universal gravitational constant. The gravitational constant, denoted G, is a Physical constant involved in the calculation of the gravitational attraction between objects with mass The above statement may be reformulated in the following way: if g is the acceleration of a reference mass at a given location in a gravitational field, then the gravitational force on an object with gravitational mass M is

This is the basis by which masses are determined by weighing. The concept of scale is applicable if a system is represented proportionally by another system In simple bathroom scales, for example, the force f is proportional to the displacement of the spring beneath the weighing pan (see Hooke's law), and the scales are calibrated to take g into account, allowing the mass M to be read off. A weighing scale (usually just "scale" in common usage except in Australian English where "scales" is more common is a Measuring instrument for A spring is a flexible elastic object used to store mechanical Energy. In Mechanics, and Physics, Hooke's law of elasticity is an approximation that states that the amount by which a material body is deformed (the Calibration is the process of establishing the relationship between a measuring device and the units of measure Note that a balance (see the subheading within Weighing scale) as used in the laboratory or the health club measures gravitational mass; only the spring scale measures weight. A weighing scale (usually just "scale" in common usage except in Australian English where "scales" is more common is a Measuring instrument for

Equivalence of inertial and gravitational masses

The equivalence of inertial and gravitational masses is sometimes referred to as the Galilean equivalence principle or weak equivalence principle. The equivalence principle The most important consequence of this equivalence principle applies to freely falling objects. Suppose we have an object with inertial and gravitational masses m and M respectively. If the only force acting on the object comes from a gravitational field g, combining Newton's second law and the gravitational law yields the acceleration

This says that the ratio of gravitational to inertial mass of any object is equal to some constant K if and only if all objects fall at the same rate in a given gravitational field. ↔ This phenomenon is referred to as the universality of free-fall. (In addition, the constant K can be taken to be 1 by defining our units appropriately. )

The first experiments demonstrating the universality of free-fall were conducted by Galileo. Galileo Galilei (15 February 1564 &ndash 8 January 1642 was a Tuscan ( Italian) Physicist, Mathematician, Astronomer, and Philosopher It is commonly stated that Galileo obtained his results by dropping objects from the Leaning Tower of Pisa, but this is most likely apocryphal; actually, he performed his experiments with balls rolling down inclined planes. The Leaning Tower of Pisa (Torre pendente di Pisa or simply The Tower of Pisa (it La Torre di Pisa is the Campanile, or freestanding bell tower of the This article deals with the physical structure For related terms see Canal inclined plane, Cable railway, Funicular, or Fixed-wing Increasingly precise experiments have been performed, such as those performed by Loránd Eötvös, using the torsion balance pendulum, in 1889. Baron Loránd von Eötvös, more commonly called Baron Roland von Eötvös in the English literature (in Hungarian Vásárosnaményi Báró Eötvös Loránd, or A torsion spring is a spring that works by torsion or twisting that is a flexible elastic object that stores Mechanical energy when it is twisted Year 1889 ( MDCCCLXXXIX) was a Common year starting on Tuesday (link will display the full calendar of the Gregorian calendar (or a Common As of 2008, no deviation from universality, and thus from Galilean equivalence, has ever been found, at least to the accuracy 1/10¹². 2008 ( MMVIII) is the current year in accordance with the Gregorian calendar, a Leap year that started on Tuesday of the Common More precise experimental efforts are still being carried out.

The universality of free-fall only applies to systems in which gravity is the only acting force. All other forces, especially friction and air resistance, must be absent or at least negligible. Friction is the Force resisting the relative motion of two Surfaces in contact or a surface in contact with a fluid (e In Fluid dynamics, drag (sometimes called fluid resistance) is the force that resists the movement of a Solid object through a Fluid (a In Engineering, Mathematics, Physics and similar disciplines the term negligible refers to the quantities so small that they can be ignored (neglected For example, if a hammer and a feather are dropped from the same height on Earth, the feather will take much longer to reach the ground; the feather is not really in free-fall because the force of air resistance upwards against the feather is comparable to the downward force of gravity. On the other hand, if the experiment is performed in a vacuum, in which there is no air resistance, the hammer and the feather should hit the ground at exactly the same time (assuming the acceleration of both objects towards each other, and of the ground towards both objects, for its own part, is negligible). This vacuum means "absence of matter" or "an empty area or space" for the cleaning appliance see Vacuum cleaner. This demonstration is easily done in a high-school laboratory, using two transparent tubes connected to a vacuum pump.

A stronger version of the equivalence principle, known as the Einstein equivalence principle or the strong equivalence principle, lies at the heart of the general theory of relativity. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 Einstein's equivalence principle states that within sufficiently small regions of space-time, it is impossible to distinguish between a uniform acceleration and a uniform gravitational field. Thus, the theory postulates that inertial and gravitational masses are fundamentally the same thing.

See also

• Weight
• Density
• Higgs boson
• Mass in special relativity
• Mass in general relativity
• Orders of magnitude (mass)
• Planck units
• Volume

References

External links

• Usenet Physics FAQ
  • Does mass change with velocity?
  • Does light have mass?
• The Origin of Mass and the Feebleness of Gravity (video) - a colloquium lecture by the Nobel Laureate Frank Wilczek
• Mass conversions
• Mass & energy
• Photons, Clocks, Gravity and the Concept of Mass by L.B.Okun
• The Apollo 15 Hammer-Feather Drop
• Apollo 15 Hammer-Feather gravity demonstration video (higher quality)
• Online mass units conversion
• Scientific American Magazine (July 2005 Issue) The Mysteries of Mass

Frank Anthony Wilczek (born May 15, 1951) is an American theoretical physicist and Nobel laureate.

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