Mechanics of planar particle motion
This article describes a particle in planar motion when observed from non-inertial reference frames. The most famous examples of planar motion are related to the motion of two spheres that are gravitationally attracted to one another, the generalization of this problem to planetary motion. See centrifugal force, two-body problem and Kepler's laws of planetary motion; those problems fall in the general field of analytical dynamics, the determination of orbits from given laws of force. This article is focused more on the kinematical issues surrounding planar motion, that is, determination of the forces necessary to result in a certain trajectory given the particle trajectory. General results presented in fictitious forces here are applied to observations of a moving particle as seen from several specific non-inertial frames, for example, a local frame, a co-rotating frame; the Lagrangian approach to fictitious forces is introduced. Unlike real forces such as electromagnetic forces, fictitious forces do not originate from physical interactions between objects.
The appearance of fictitious forces is associated with use of a non-inertial frame of reference, their absence with use of an inertial frame of reference. The connection between inertial frames and fictitious forces, is expressed, for example, by Arnol'd: The equations of motion in a non-inertial system differ from the equations in an inertial system by additional terms called inertial forces; this allows us to detect experimentally the non-inertial nature of a system. A different tack on the subject is provided by Iro: An additional force due to nonuniform relative motion of two reference frames is called a pseudo-force. Fictitious forces do not appear in the equations of motion in an inertial frame of reference: in an inertial frame, the motion of an object is explained by the real impressed forces. In a non-inertial frame such as a rotating frame, Newton's first and second laws still can be used to make accurate physical predictions provided fictitious forces are included along with the real forces.
For solving problems of mechanics in non-inertial reference frames, the advice given in textbooks is to treat the fictitious forces like real forces and to pretend you are in an inertial frame. Treat the fictitious forces like real forces, pretend you are in an inertial frame, it should be mentioned that "treating the fictitious forces like real forces" means, in particular, that fictitious forces as seen in a particular non-inertial frame transform as vectors under coordinate transformations made within that frame, that is, like real forces. Next, it is observed that time varying coordinates are used in both inertial and non-inertial frames of reference, so the use of time varying coordinates should not be confounded with a change of observer, but is only a change of the observer's choice of description. Elaboration of this point and some citations on the subject follow; the term frame of reference is used in a broad sense, but for the present discussion its meaning is restricted to refer to an observer's state of motion, that is, to either an inertial frame of reference or a non-inertial frame of reference.
The term coordinate system is used to differentiate between different possible choices for a set of variables to describe motion, choices available to any observer, regardless of their state of motion. Examples are Cartesian coordinates, polar coordinates and curvilinear coordinates. Here are two quotes relating "state of motion" and "coordinate system": We first introduce the notion of reference frame, itself related to the idea of observer: the reference frame is, in some sense, the "Euclidean space carried by the observer". Let us give a more mathematical definition:… the reference frame is... the set of all points in the Euclidean space with the rigid body motion of the observer. The frame, denoted R, is said to move with the observer.… The spatial positions of particles are labelled relative to a frame R by establishing a coordinate system R with origin O. The corresponding set of axes, sharing the rigid body motion of the frame R, can be considered to give a physical realization of R.
In a frame R, coordinates are changed from R to R' by carrying out, at each instant of time, the same coordinate transformation on the components of intrinsic objects introduced to represent physical quantities in this frame. In traditional developments of special and general relativity it has been customary not to distinguish between two quite distinct ideas; the first is the notion of a coordinate system, understood as the smooth, invertible assignment of four numbers to events in spacetime neighborhoods. The second, the frame of reference, refers to an idealized system used to assign such numbers … To avoid unnecessary restrictions, we can divorce this arrangement from metrical notions. … Of special importance for our purposes is that each frame of reference has a definite state of motion at each event of spacetime.…Within the context of special relativity and as long as we restrict ourselves to frames of reference in inertial motion little of importance depends on the difference between an inertial frame of reference and the inertial coordinate system it induces.
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Space is the boundless three-dimensional extent in which objects and events have relative position and direction. Physical space is conceived in three linear dimensions, although modern physicists consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime; the concept of space is considered to be of fundamental importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework. Debates concerning the nature and the mode of existence of space date back to antiquity. Many of these classical philosophical questions were discussed in the Renaissance and reformulated in the 17th century during the early development of classical mechanics. In Isaac Newton's view, space was absolute—in the sense that it existed permanently and independently of whether there was any matter in the space. Other natural philosophers, notably Gottfried Leibniz, thought instead that space was in fact a collection of relations between objects, given by their distance and direction from one another.
In the 18th century, the philosopher and theologian George Berkeley attempted to refute the "visibility of spatial depth" in his Essay Towards a New Theory of Vision. The metaphysician Immanuel Kant said that the concepts of space and time are not empirical ones derived from experiences of the outside world—they are elements of an given systematic framework that humans possess and use to structure all experiences. Kant referred to the experience of "space" in his Critique of Pure Reason as being a subjective "pure a priori form of intuition". In the 19th and 20th centuries mathematicians began to examine geometries that are non-Euclidean, in which space is conceived as curved, rather than flat. According to Albert Einstein's theory of general relativity, space around gravitational fields deviates from Euclidean space. Experimental tests of general relativity have confirmed that non-Euclidean geometries provide a better model for the shape of space. Galilean and Cartesian theories about space and motion are at the foundation of the Scientific Revolution, understood to have culminated with the publication of Newton's Principia in 1687.
Newton's theories about space and time helped. While his theory of space is considered the most influential in Physics, it emerged from his predecessors' ideas about the same; as one of the pioneers of modern science, Galilei revised the established Aristotelian and Ptolemaic ideas about a geocentric cosmos. He backed the Copernican theory that the universe was heliocentric, with a stationary sun at the center and the planets—including the Earth—revolving around the sun. If the Earth moved, the Aristotelian belief that its natural tendency was to remain at rest was in question. Galilei wanted to prove instead that the sun moved around its axis, that motion was as natural to an object as the state of rest. In other words, for Galilei, celestial bodies, including the Earth, were inclined to move in circles; this view displaced another Aristotelian idea—that all objects gravitated towards their designated natural place-of-belonging. Descartes set out to replace the Aristotelian worldview with a theory about space and motion as determined by natural laws.
In other words, he sought a metaphysical foundation or a mechanical explanation for his theories about matter and motion. Cartesian space was Euclidean in structure—infinite and flat, it was defined as that. The Cartesian notion of space is linked to his theories about the nature of the body and matter, he is famously known for his "cogito ergo sum", or the idea that we can only be certain of the fact that we can doubt, therefore think and therefore exist. His theories belong to the rationalist tradition, which attributes knowledge about the world to our ability to think rather than to our experiences, as the empiricists believe, he posited a clear distinction between the body and mind, referred to as the Cartesian dualism. Following Galilei and Descartes, during the seventeenth century the philosophy of space and time revolved around the ideas of Gottfried Leibniz, a German philosopher–mathematician, Isaac Newton, who set out two opposing theories of what space is. Rather than being an entity that independently exists over and above other matter, Leibniz held that space is no more than the collection of spatial relations between objects in the world: "space is that which results from places taken together".
Unoccupied regions are those that could have objects in them, thus spatial relations with other places. For Leibniz space was an idealised abstraction from the relations between individual entities or their possible locations and therefore could not be continuous but must be discrete. Space could be thought of in a similar way to the relations between family members. Although people in the family are related to one another, the relations do not exist independently of the people. Leibniz argued that space could not exist independently of objects in the world because that implies a difference between two universes alike except for the location of the material world in
In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes; the same amount of work is done by the body when decelerating from its current speed to a state of rest. In classical mechanics, the kinetic energy of a non-rotating object of mass m traveling at a speed v is 1 2 m v 2. In relativistic mechanics, this is a good approximation only when v is much less than the speed of light; the standard unit of kinetic energy is the joule. The imperial unit of kinetic energy is the foot-pound; the adjective kinetic has its roots in the Greek word κίνησις kinesis, meaning "motion". The dichotomy between kinetic energy and potential energy can be traced back to Aristotle's concepts of actuality and potentiality; the principle in classical mechanics that E ∝ mv2 was first developed by Gottfried Leibniz and Johann Bernoulli, who described kinetic energy as the living force, vis viva.
Willem's Gravesande of the Netherlands provided experimental evidence of this relationship. By dropping weights from different heights into a block of clay, Willem's Gravesande determined that their penetration depth was proportional to the square of their impact speed. Émilie du Châtelet published an explanation. The terms kinetic energy and work in their present scientific meanings date back to the mid-19th century. Early understandings of these ideas can be attributed to Gaspard-Gustave Coriolis, who in 1829 published the paper titled Du Calcul de l'Effet des Machines outlining the mathematics of kinetic energy. William Thomson Lord Kelvin, is given the credit for coining the term "kinetic energy" c. 1849–51. Energy occurs in many forms, including chemical energy, thermal energy, electromagnetic radiation, gravitational energy, electric energy, elastic energy, nuclear energy, rest energy; these can be categorized in two main classes: kinetic energy. Kinetic energy is the movement energy of an object.
Kinetic energy can be transformed into other kinds of energy. Kinetic energy may be best understood by examples that demonstrate how it is transformed to and from other forms of energy. For example, a cyclist uses chemical energy provided by food to accelerate a bicycle to a chosen speed. On a level surface, this speed can be maintained without further work, except to overcome air resistance and friction; the chemical energy has been converted into kinetic energy, the energy of motion, but the process is not efficient and produces heat within the cyclist. The kinetic energy in the moving cyclist and the bicycle can be converted to other forms. For example, the cyclist could encounter a hill just high enough to coast up, so that the bicycle comes to a complete halt at the top; the kinetic energy has now been converted to gravitational potential energy that can be released by freewheeling down the other side of the hill. Since the bicycle lost some of its energy to friction, it never regains all of its speed without additional pedaling.
The energy is not destroyed. Alternatively, the cyclist could connect a dynamo to one of the wheels and generate some electrical energy on the descent; the bicycle would be traveling slower at the bottom of the hill than without the generator because some of the energy has been diverted into electrical energy. Another possibility would be for the cyclist to apply the brakes, in which case the kinetic energy would be dissipated through friction as heat. Like any physical quantity, a function of velocity, the kinetic energy of an object depends on the relationship between the object and the observer's frame of reference. Thus, the kinetic energy of an object is not invariant. Spacecraft use chemical energy to launch and gain considerable kinetic energy to reach orbital velocity. In an circular orbit, this kinetic energy remains constant because there is no friction in near-earth space. However, it becomes apparent at re-entry. If the orbit is elliptical or hyperbolic throughout the orbit kinetic and potential energy are exchanged.
Without loss or gain, the sum of the kinetic and potential energy remains constant. Kinetic energy can be passed from one object to another. In the game of billiards, the player imposes kinetic energy on the cue ball by striking it with the cue stick. If the cue ball collides with another ball, it slows down and the ball it hit accelerates its speed as the kinetic energy is passed on to it. Collisions in billiards are elastic collisions, in which kinetic energy is preserved. In inelastic collisions, kinetic energy is dissipated in various forms of energy, such as heat, binding energy. Flywheels have been developed as a method of energy storage; this illustrates that kinetic energy is stored in rotational motion. Several mathematical descriptions of kinetic energy exist that describe it in the appropriate physical situation. For objects and processes in common human experience, the formula ½mv² given by Newtonian mechanics is suitable. However, if the speed of the object is comparabl
Friction is the force resisting the relative motion of solid surfaces, fluid layers, material elements sliding against each other. There are several types of friction: Dry friction is a force that opposes the relative lateral motion of two solid surfaces in contact. Dry friction is subdivided into static friction between non-moving surfaces, kinetic friction between moving surfaces. With the exception of atomic or molecular friction, dry friction arises from the interaction of surface features, known as asperities Fluid friction describes the friction between layers of a viscous fluid that are moving relative to each other. Lubricated friction is a case of fluid friction where a lubricant fluid separates two solid surfaces. Skin friction is a component of drag, the force resisting the motion of a fluid across the surface of a body. Internal friction is the force resisting motion between the elements making up a solid material while it undergoes deformation; when surfaces in contact move relative to each other, the friction between the two surfaces converts kinetic energy into thermal energy.
This property can have dramatic consequences, as illustrated by the use of friction created by rubbing pieces of wood together to start a fire. Kinetic energy is converted to thermal energy whenever motion with friction occurs, for example when a viscous fluid is stirred. Another important consequence of many types of friction can be wear, which may lead to performance degradation or damage to components. Friction is a component of the science of tribology. Friction is important in supplying traction to facilitate motion on land. Most land vehicles rely on friction for acceleration and changing direction. Sudden reductions in traction can cause loss of control and accidents. Friction is not itself a fundamental force. Dry friction arises from a combination of inter-surface adhesion, surface roughness, surface deformation, surface contamination; the complexity of these interactions makes the calculation of friction from first principles impractical and necessitates the use of empirical methods for analysis and the development of theory.
Friction is a non-conservative force - work done against friction is path dependent. In the presence of friction, some energy is always lost in the form of heat, thus mechanical energy is not conserved. The Greeks, including Aristotle and Pliny the Elder, were interested in the cause and mitigation of friction, they were aware of differences between static and kinetic friction with Themistius stating in 350 A. D. that "it is easier to further the motion of a moving body than to move a body at rest". The classic laws of sliding friction were discovered by Leonardo da Vinci in 1493, a pioneer in tribology, but the laws documented in his notebooks, were not published and remained unknown; these laws were rediscovered by Guillaume Amontons in 1699 and became known as Amonton's three laws of dry friction. Amontons presented the nature of friction in terms of surface irregularities and the force required to raise the weight pressing the surfaces together; this view was further elaborated by Bernard Forest de Bélidor and Leonhard Euler, who derived the angle of repose of a weight on an inclined plane and first distinguished between static and kinetic friction.
John Theophilus Desaguliers first recognized the role of adhesion in friction. Microscopic forces cause surfaces to stick together; the understanding of friction was further developed by Charles-Augustin de Coulomb. Coulomb investigated the influence of four main factors on friction: the nature of the materials in contact and their surface coatings. Coulomb further considered the influence of sliding velocity and humidity, in order to decide between the different explanations on the nature of friction, proposed; the distinction between static and dynamic friction is made in Coulomb's friction law, although this distinction was drawn by Johann Andreas von Segner in 1758. The effect of the time of repose was explained by Pieter van Musschenbroek by considering the surfaces of fibrous materials, with fibers meshing together, which takes a finite time in which the friction increases. John Leslie noted a weakness in the views of Amontons and Coulomb: If friction arises from a weight being drawn up the inclined plane of successive asperities, why isn't it balanced through descending the opposite slope?
Leslie was skeptical about the role of adhesion proposed by Desaguliers, which should on the whole have the same tendency to accelerate as to retard the motion. In Leslie's view, friction should be seen as a time-dependent process of flattening, pressing down asperities, which creates new obstacles in what were cavities before. Arthur Jules Morin developed the concept of sliding versus rolling friction. Osborne Reynolds derived the equation of viscous flow; this completed the classic empirical model of friction used today in engineering. In 1877, Fleeming Jenkin and J. A. Ewing investigated the continuity between static and kinetic friction; the focus of research during the 20th century has been to understand the physical mechanisms behind friction. Frank Philip Bowden and David Tabor showed that, at a microscopic level, the actual area of contact between surfaces is a small fraction of the apparent area; this actual area of contact, caused by asperities increases with pressure. The development of the atomic force microscope
In physics, a rigid body is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external forces exerted on it. A rigid body is considered as a continuous distribution of mass. In the study of special relativity, a rigid body does not exist. In quantum mechanics a rigid body is thought of as a collection of point masses. For instance, in quantum mechanics molecules are seen as rigid bodies; the position of a rigid body is the position of all the particles. To simplify the description of this position, we exploit the property that the body is rigid, namely that all its particles maintain the same distance relative to each other. If the body is rigid, it is sufficient to describe the position of at least three non-collinear particles; this makes it possible to reconstruct the position of all the other particles, provided that their time-invariant position relative to the three selected particles is known.
However a different, mathematically more convenient, but equivalent approach is used. The position of the whole body is represented by: the linear position or position of the body, namely the position of one of the particles of the body chosen as a reference point, together with the angular position of the body. Thus, the position of a rigid body has two components: angular, respectively; the same is true for other kinematic and kinetic quantities describing the motion of a rigid body, such as linear and angular velocity, momentum and kinetic energy. The linear position can be represented by a vector with its tail at an arbitrary reference point in space and its tip at an arbitrary point of interest on the rigid body coinciding with its center of mass or centroid; this reference point may define the origin of a coordinate system fixed to the body. There are several ways to numerically describe the orientation of a rigid body, including a set of three Euler angles, a quaternion, or a direction cosine matrix.
All these methods define the orientation of a basis set which has a fixed orientation relative to the body, relative to another basis set, from which the motion of the rigid body is observed. For instance, a basis set with fixed orientation relative to an airplane can be defined as a set of three orthogonal unit vectors b1, b2, b3, such that b1 is parallel to the chord line of the wing and directed forward, b2 is normal to the plane of symmetry and directed rightward, b3 is given by the cross product b 3 = b 1 × b 2. In general, when a rigid body moves, both its position and orientation vary with time. In the kinematic sense, these changes are referred to as rotation, respectively. Indeed, the position of a rigid body can be viewed as a hypothetic translation and rotation of the body starting from a hypothetic reference position. Velocity and angular velocity are measured with respect to a frame of reference; the linear velocity of a rigid body is a vector quantity, equal to the time rate of change of its linear position.
Thus, it is the velocity of a reference point fixed to the body. During purely translational motion, all points on a rigid body move with the same velocity. However, when motion involves rotation, the instantaneous velocity of any two points on the body will not be the same. Two points of a rotating body will have the same instantaneous velocity only if they happen to lie on an axis parallel to the instantaneous axis of rotation. Angular velocity is a vector quantity that describes the angular speed at which the orientation of the rigid body is changing and the instantaneous axis about which it is rotating. All points on a rigid body experience the same angular velocity at all times. During purely rotational motion, all points on the body change position except for those lying on the instantaneous axis of rotation; the relationship between orientation and angular velocity is not directly analogous to the relationship between position and velocity. Angular velocity is not the time rate of change of orientation, because there is no such concept as an orientation vector that can be differentiated to obtain the angular velocity.
The angular velocity of a rigid body B in a reference frame N is equal to the sum of the angular velocity of a rigid body D in N and the angular velocity of B with respect to D: N ω B = N ω D + D ω B. In this case, rigid bodies and reference frames are indistinguishable and interchangeable. For any set of three points P, Q, R, the position ve
Virtual work arises in the application of the principle of least action to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement is different for different displacements. Among all the possible displacements that a particle may follow, called virtual displacements, one will minimize the action; this displacement is therefore the displacement followed by the particle according to the principle of least action. The work of a force on a particle along a virtual displacement is known as the virtual work. Virtual work and the associated calculus of variations were formulated to analyze systems of rigid bodies, but they have been developed for the study of the mechanics of deformable bodies; the principle of virtual work had always been used in some form since antiquity in the study of statics. It was used by the Greeks, medieval Arabs and Latins, Renaissance Italians as "the law of lever"; the idea of virtual work was invoked by many notable physicists of the 17th century, such as Galileo, Torricelli and Huygens, in varying degrees of generality, when solving problems in statics.
Working with Leibnizian concepts, Johann Bernoulli systematized the virtual work principle and made explicit the concept of infinitesimal displacement. He was able to solve problems for both rigid bodies as well as fluids. Bernoulli's version of virtual work law appeared in his letter to Pierre Varignon in 1715, published in Varignon's second volume of Nouvelle mécanique ou Statique in 1725; this formulation of the principle is today known as the principle of virtual velocities and is considered as the prototype of the contemporary virtual work principles. In 1743 D'Alembert published his Traité de Dynamique where he applied the principle of virtual work, based on Bernoulli's work, to solve various problems in dynamics, his idea was to convert a dynamical problem into static problem by introducing inertial force. In 1768, Lagrange presented the virtual work principle in a more efficient form by introducing generalized coordinates and presented it as an alternative principle of mechanics by which all problems of equilibrium could be solved.
A systematic exposition of Lagrange's program of applying this approach to all of mechanics, both static and dynamic D'Alembert's principle, was given in his Mécanique Analytique of 1788. Although Lagrange had presented his version of least action principle prior to this work, he recognized the virtual work principle to be more fundamental because it could be assumed alone as the foundation for all mechanics, unlike the modern understanding that least action does not account for non-conservative forces. If a force acts on a particle as it moves from point A to point B for each possible trajectory that the particle may take, it is possible to compute the total work done by the force along the path; the principle of virtual work, the form of the principle of least action applied to these systems, states that the path followed by the particle is the one for which the difference between the work along this path and other nearby paths is zero. The formal procedure for computing the difference of functions evaluated on nearby paths is a generalization of the derivative known from differential calculus, is termed the calculus of variations.
Consider a point particle that moves along a path, described by a function r from point A, where r, to point B, where r. It is possible that the particle moves from A to B along a nearby path described by r + δr, where δr is called the variation of r; the variation δr satisfies the requirement δr = δr = 0. The components of the variation, δr1, δr2 and δr3, are called virtual displacements; this can be generalized to an arbitrary mechanical system defined by the generalized coordinates qi, i = 1... n. In which case, the variation of the trajectory qi is defined by the virtual displacements δqi, i = 1... n. Virtual work is the total work done by the applied forces and the inertial forces of a mechanical system as it moves through a set of virtual displacements; when considering forces applied to a body in static equilibrium, the principle of least action requires the virtual work of these forces to be zero. Consider a particle P that moves from a point A to a point B along a trajectory r, while a force F is applied to it.
The work done by the force F is given by the integral W = ∫ r = A r = B F ⋅ d r = ∫ t 0 t 1 F ⋅ d r d t d t = ∫ t 0 t 1 F ⋅ v d t, where dr is the differential element along the curve, the trajectory of P, v is its velocity. It is important to notice. Now consider particle P that moves from point A to point B again, but this time it moves along the nearby trajectory that differs from r by the variation δr=εh, where ε is a scaling constant that can be made as small as desired and h(
Mass is both a property of a physical body and a measure of its resistance to acceleration when a net force is applied. The object's mass determines the strength of its gravitational attraction to other bodies; the basic SI unit of mass is the kilogram. In physics, mass is not the same as weight though mass is determined by measuring the object's weight using a spring scale, rather than balance scale comparing it directly with known masses. An object on the Moon would weigh less than it does on Earth because of the lower gravity, but it would still have the same mass; this is because weight is a force, while mass is the property that determines the strength of this force. There are several distinct phenomena. Although some theorists have speculated that some of these phenomena could be independent of each other, current experiments have found no difference in results regardless of how it is measured: Inertial mass measures an object's resistance to being accelerated by a force. Active gravitational mass measures the gravitational force exerted by an object.
Passive gravitational mass measures the gravitational force exerted on an object in a known gravitational field. The mass of an object determines its acceleration in the presence of an applied force; the inertia and the inertial mass describe the same properties of physical bodies at the qualitative and quantitative level by other words, the mass quantitatively describes the inertia. According to Newton's second law of motion, if a body of fixed mass m is subjected to a single force F, its acceleration a is given by F/m. A body's mass determines the degree to which it generates or is affected by a gravitational field. If a first body of mass mA is placed at a distance r from a second body of mass mB, each body is subject to an attractive force Fg = GmAmB/r2, where G = 6.67×10−11 N kg−2 m2 is the "universal gravitational constant". This is sometimes referred to as gravitational mass. Repeated experiments since the 17th century have demonstrated that inertial and gravitational mass are identical.
The standard International System of Units unit of mass is the kilogram. The kilogram is 1000 grams, first defined in 1795 as one cubic decimeter of water at the melting point of ice. However, because precise measurement of a decimeter of water at the proper temperature and pressure was difficult, in 1889 the kilogram was redefined as the mass of the international prototype kilogram of cast iron, thus became independent of the meter and the properties of water. However, the mass of the international prototype and its identical national copies have been found to be drifting over time, it is expected that the re-definition of the kilogram and several other units will occur on May 20, 2019, following a final vote by the CGPM in November 2018. The new definition will use only invariant quantities of nature: the speed of light, the caesium hyperfine frequency, the Planck constant. Other units are accepted for use in SI: the tonne is equal to 1000 kg. the electronvolt is a unit of energy, but because of the mass–energy equivalence it can be converted to a unit of mass, is used like one.
In this context, the mass has units of eV/c2. The electronvolt and its multiples, such as the MeV, are used in particle physics; the atomic mass unit is 1/12 of the mass of a carbon-12 atom 1.66×10−27 kg. The atomic mass unit is convenient for expressing the masses of molecules. Outside the SI system, other units of mass include: the slug is an Imperial unit of mass; the pound is a unit of both mass and force, used in the United States. In scientific contexts where pound and pound need to be distinguished, SI units are used instead; the Planck mass is the maximum mass of point particles. It is used in particle physics; the solar mass is defined as the mass of the Sun. It is used in astronomy to compare large masses such as stars or galaxies; the mass of a small particle may be identified by its inverse Compton wavelength. The mass of a large star or black hole may be identified with its Schwarzschild radius. In physical science, one may distinguish conceptually between at least seven different aspects of mass, or seven physical notions that involve the concept of mass.
Every experiment to date has shown these seven values to be proportional, in some cases equal, this proportionality gives rise to the abstract concept of mass. There are a number of ways mass can be measured or operationally defined: Inertial mass is a measure of an object's resistance to acceleration when a force is applied, it is determined by applying a force to an object and measuring the acceleration that results from that force. An object with small inertial mass will accelerate more than an object with large inertial mass when acted upon by the same force. One says. Active gravitational mass is a measure of the strength of an object's gravitational flux. Gravitational field can be measured by allowing a small "test object" to fall and measuring its free-fall acceleration. For example, an object in free fall near the Moon is subject to a smaller gravitational field, hence