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(
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potential energy of an object that depends on its mass and its distance from the center of mass of another object, the elastic potential energy of an extended spring, the electric potential energy of an electric charge in an electric field; the unit for energy in the International System of Units is the joule, which has the symbol J. The term potential energy was introduced by the 19th-century Scottish engineer and physicist William Rankine, although it has links to Greek philosopher Aristotle's concept of potentiality. Potential energy is associated with forces that act on a body in a way that the total work done by these forces on the body depends only on the initial and final positions of the body in space; these forces, that are called conservative forces, can be represented at every point in space by vectors expressed as gradients of a certain scalar function called potential.
Since the work of potential forces acting on a body that moves from a start to an end position is determined only by these two positions, does not depend on the trajectory of the body, there is a function known as potential that can be evaluated at the two positions to determine this work. There are various types of potential energy, each associated with a particular type of force. For example, the work of an elastic force is called elastic potential energy. Chemical potential energy, such as the energy stored in fossil fuels, is the work of the Coulomb force during rearrangement of mutual positions of electrons and nuclei in atoms and molecules. Thermal energy has two components: the kinetic energy of random motions of particles and the potential energy of their mutual positions. Forces derivable from a potential are called conservative forces; the work done by a conservative force is W = − Δ U where Δ U is the change in the potential energy associated with the force. The negative sign provides the convention that work done against a force field increases potential energy, while work done by the force field decreases potential energy.
Common notations for potential energy are PE, U, V, Ep. Potential energy is the energy by virtue of an object's position relative to other objects. Potential energy is associated with restoring forces such as a spring or the force of gravity; the action of stretching a spring or lifting a mass is performed by an external force that works against the force field of the potential. This work is stored in the force field, said to be stored as potential energy. If the external force is removed the force field acts on the body to perform the work as it moves the body back to the initial position, reducing the stretch of the spring or causing a body to fall. Consider a ball whose mass is m and whose height is h; the acceleration g of free fall is constant, so the weight force of the ball mg is constant. Force × displacement gives the work done, equal to the gravitational potential energy, thus U g = m g h The more formal definition is that potential energy is the energy difference between the energy of an object in a given position and its energy at a reference position.
Potential energy is linked with forces. If the work done by a force on a body that moves from A to B does not depend on the path between these points the work of this force measured from A assigns a scalar value to every other point in space and defines a scalar potential field. In this case, the force can be defined as the negative of the vector gradient of the potential field. If the work for an applied force is independent of the path the work done by the force is evaluated at the start and end of the trajectory of the point of application; this means that there is a function U, called a "potential," that can be evaluated at the two points xA and xB to obtain the work over any trajectory between these two points. It is tradition to define this function with a negative sign so that positive work is a reduction in the potential, W = ∫ C F ⋅ d x = U − U where C is the trajectory taken from A to B; because the work done is independent of the path taken this expression is true for any trajectory, C, from A to B.
The function U is called the potential energy associated with the applied force. Examples of forces that have potential energies are spring forces. In this section the relationship between work and potential energy is presented in more detail; the line integral that defines work along curve C takes a special form if the force F is related to a scalar field φ so that F = ∇ φ = ( ∂ φ ∂ x, ∂
Moment of inertia
The moment of inertia, otherwise known as the angular mass or rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rotation rate, it is an extensive property: for a point mass the moment of inertia is just the mass times the square of the perpendicular distance to the rotation axis. The moment of inertia of a rigid composite system is the sum of the moments of inertia of its component subsystems, its simplest definition is the second moment of mass with respect to distance from an axis. For bodies constrained to rotate in a plane, only their moment of inertia about an axis perpendicular to the plane, a scalar value, matters. For bodies free to rotate in three dimensions, their moments can be described by a symmetric 3 × 3 matrix, with a set of mutually perpendicular principal axes for which this matrix is diagonal and torques around the axes act independently of each other.
When a body is free to rotate around an axis, torque must be applied to change its angular momentum. The amount of torque needed to cause any given angular acceleration is proportional to the moment of inertia of the body. Moment of inertia may be expressed in units of kilogram meter squared in SI units and pound-foot-second squared in imperial or US units. Moment of inertia plays the role in rotational kinetics that mass plays in linear kinetics - both characterize the resistance of a body to changes in its motion; the moment of inertia depends on how mass is distributed around an axis of rotation, will vary depending on the chosen axis. For a point-like mass, the moment of inertia about some axis is given by m r 2, where r is the distance of the point from the axis, m is the mass. For an extended rigid body, the moment of inertia is just the sum of all the small pieces of mass multiplied by the square of their distances from the axis in question. For an extended body of a regular shape and uniform density, this summation sometimes produces a simple expression that depends on the dimensions and total mass of the object.
In 1673 Christiaan Huygens introduced this parameter in his study of the oscillation of a body hanging from a pivot, known as a compound pendulum. The term moment of inertia was introduced by Leonhard Euler in his book Theoria motus corporum solidorum seu rigidorum in 1765, it is incorporated into Euler's second law; the natural frequency of oscillation of a compound pendulum is obtained from the ratio of the torque imposed by gravity on the mass of the pendulum to the resistance to acceleration defined by the moment of inertia. Comparison of this natural frequency to that of a simple pendulum consisting of a single point of mass provides a mathematical formulation for moment of inertia of an extended body. Moment of inertia appears in momentum, kinetic energy, in Newton's laws of motion for a rigid body as a physical parameter that combines its shape and mass. There is an interesting difference in the way moment of inertia appears in planar and spatial movement. Planar movement has a single scalar that defines the moment of inertia, while for spatial movement the same calculations yield a 3 × 3 matrix of moments of inertia, called the inertia matrix or inertia tensor.
The moment of inertia of a rotating flywheel is used in a machine to resist variations in applied torque to smooth its rotational output. The moment of inertia of an airplane about its longitudinal and vertical axis determines how steering forces on the control surfaces of its wings and tail affect the plane in roll and yaw. Moment of inertia I is defined as the ratio of the net angular momentum L of a system to its angular velocity ω around a principal axis, I = L ω. If the angular momentum of a system is constant as the moment of inertia gets smaller, the angular velocity must increase; this occurs when spinning figure skaters pull in their outstretched arms or divers curl their bodies into a tuck position during a dive, to spin faster. If the shape of the body does not change its moment of inertia appears in Newton's law of motion as the ratio of an applied torque τ on a body to the angular acceleration α around a principal axis, τ = I α. For a simple pendulum, this definition yields a formula for the moment of inertia I in terms of the mass m of the pendulum and its distance r from the pivot point as, I = m r 2.
Thus, moment of inertia depends on both the mass m of a body and its geometry, or shape, as defined by the distance r to the axis of rotation. This simple formula generalizes to define moment of inertia for an arbitrarily shaped body as the sum of all the elemental point masses d m each multiplied by the square of its perpendicular distance
In Newtonian mechanics, linear momentum, translational momentum, or momentum is the product of the mass and velocity of an object. It is a vector quantity, possessing a direction in three-dimensional space. If m is an object's mass and v is the velocity the momentum is p = m v, In SI units, it is measured in kilogram meters per second. Newton's second law of motion states that a body's rate of change in momentum is equal to the net force acting on it. Momentum depends on the frame of reference, but in any inertial frame it is a conserved quantity, meaning that if a closed system is not affected by external forces, its total linear momentum does not change. Momentum is conserved in special relativity and, in a modified form, in electrodynamics, quantum mechanics, quantum field theory, general relativity, it is an expression of one of the fundamental symmetries of time: translational symmetry. Advanced formulations of classical mechanics and Hamiltonian mechanics, allow one to choose coordinate systems that incorporate symmetries and constraints.
In these systems the conserved quantity is generalized momentum, in general this is different from the kinetic momentum defined above. The concept of generalized momentum is carried over into quantum mechanics, where it becomes an operator on a wave function; the momentum and position operators are related by the Heisenberg uncertainty principle. In continuous systems such as electromagnetic fields and deformable bodies, a momentum density can be defined, a continuum version of the conservation of momentum leads to equations such as the Navier–Stokes equations for fluids or the Cauchy momentum equation for deformable solids or fluids. Momentum is a vector quantity: it has both magnitude and direction. Since momentum has a direction, it can be used to predict the resulting direction and speed of motion of objects after they collide. Below, the basic properties of momentum are described in one dimension; the vector equations are identical to the scalar equations. The momentum of a particle is conventionally represented by the letter p.
It is the product of two quantities, the particle's mass and its velocity: p = m v. The unit of momentum is the product of the units of velocity. In SI units, if the mass is in kilograms and the velocity is in meters per second the momentum is in kilogram meters per second. In cgs units, if the mass is in grams and the velocity in centimeters per second the momentum is in gram centimeters per second. Being a vector, momentum has direction. For example, a 1 kg model airplane, traveling due north at 1 m/s in straight and level flight, has a momentum of 1 kg⋅m/s due north measured with reference to the ground; the momentum of a system of particles is the vector sum of their momenta. If two particles have respective masses m1 and m2, velocities v1 and v2, the total momentum is p = p 1 + p 2 = m 1 v 1 + m 2 v 2; the momenta of more than two particles can be added more with the following: p = ∑ i m i v i. A system of particles has a center of mass, a point determined by the weighted sum of their positions: r cm = m 1 r 1 + m 2 r 2 + ⋯ m 1 + m 2 + ⋯ = ∑ i m i r i ∑ i m i.
If all the particles are moving, the center of mass will be moving as well. If the center of mass is moving at velocity vcm, the momentum is: p = m v cm; this is known as Euler's first law. If the net force applied to a particle is a constant F, is applied for a time interval Δt, the momentum of the particle changes by an amount Δ p = F Δ t. In differential form, this is Newton's second law. If the net force experienced by a particle changes as a function of time, F, the change in momentum between times t1 and t2 is Δ p = J = ∫ t 1
In everyday use and in kinematics, the speed of an object is the magnitude of its velocity. The average speed of an object in an interval of time is the distance travelled by the object divided by the duration of the interval. Speed has the dimensions of distance divided by time; the SI unit of speed is the metre per second, but the most common unit of speed in everyday usage is the kilometre per hour or, in the US and the UK, miles per hour. For air and marine travel the knot is used; the fastest possible speed at which energy or information can travel, according to special relativity, is the speed of light in a vacuum c = 299792458 metres per second. Matter can not quite reach the speed of light. In relativity physics, the concept of rapidity replaces the classical idea of speed. Italian physicist Galileo Galilei is credited with being the first to measure speed by considering the distance covered and the time it takes. Galileo defined speed as the distance covered per unit of time. In equation form, v = d t, where v is speed, d is distance, t is time.
A cyclist who covers 30 metres in a time of 2 seconds, for example, has a speed of 15 metres per second. Objects in motion have variations in speed. Speed at some instant, or assumed constant during a short period of time, is called instantaneous speed. By looking at a speedometer, one can read the instantaneous speed of a car at any instant. A car travelling at 50 km/h goes for less than one hour at a constant speed, but if it did go at that speed for a full hour, it would travel 50 km. If the vehicle continued at that speed for half an hour, it would cover half that distance. If it continued for only one minute, it would cover about 833 m. In mathematical terms, the instantaneous speed v is defined as the magnitude of the instantaneous velocity v, that is, the derivative of the position r with respect to time: v = | v | = | r ˙ | = | d r d t |. If s is the length of the path travelled until time t, the speed equals the time derivative of s: v = d s d t. In the special case where the velocity is constant, this can be simplified to v = s / t.
The average speed over a finite time interval is the total distance travelled divided by the time duration. Different from instantaneous speed, average speed is defined as the total distance covered divided by the time interval. For example, if a distance of 80 kilometres is driven in 1 hour, the average speed is 80 kilometres per hour. If 320 kilometres are travelled in 4 hours, the average speed is 80 kilometres per hour; when a distance in kilometres is divided by a time in hours, the result is in kilometres per hour. Average speed does not describe the speed variations that may have taken place during shorter time intervals, so average speed is quite different from a value of instantaneous speed. If the average speed and the time of travel are known, the distance travelled can be calculated by rearranging the definition to d = v ¯ t. Using this equation for an average speed of 80 kilometres per hour on a 4-hour trip, the distance covered is found to be 320 kilometres. Expressed in graphical language, the slope of a tangent line at any point of a distance-time graph is the instantaneous speed at this point, while the slope of a chord line of the same graph is the average speed during the time interval covered by the chord.
Average speed of an object is Vav = s÷t Linear speed is the distance travelled per unit of time, while tangential speed is the linear speed of something moving along a circular path. A point on the outside edge of a merry-go-round or turntable travels a greater distance in one complete rotation than a point nearer the center. Travelling a greater distance in the same time means a greater speed, so linear speed is greater on the outer edge of a rotating object than it is closer to the axis; this speed along a circular path is known as tangential speed because the direction of motion is tangent to the circumference of the circle. For circular motion, the terms linear speed and tangential speed are used interchangeably, both use units of m/s, km/h, others. Rotational speed involves the number of revolutions per unit of time. All parts of a rigid merry-
Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, astronomical objects, such as spacecraft, planets and galaxies. If the present state of an object is known it is possible to predict by the laws of classical mechanics how it will move in the future and how it has moved in the past; the earliest development of classical mechanics is referred to as Newtonian mechanics. It consists of the physical concepts employed by and the mathematical methods invented by Isaac Newton and Gottfried Wilhelm Leibniz and others in the 17th century to describe the motion of bodies under the influence of a system of forces. More abstract methods were developed, leading to the reformulations of classical mechanics known as Lagrangian mechanics and Hamiltonian mechanics; these advances, made predominantly in the 18th and 19th centuries, extend beyond Newton's work through their use of analytical mechanics. They are, with some modification used in all areas of modern physics.
Classical mechanics provides accurate results when studying large objects that are not massive and speeds not approaching the speed of light. When the objects being examined have about the size of an atom diameter, it becomes necessary to introduce the other major sub-field of mechanics: quantum mechanics. To describe velocities that are not small compared to the speed of light, special relativity is needed. In case that objects become massive, general relativity becomes applicable. However, a number of modern sources do include relativistic mechanics into classical physics, which in their view represents classical mechanics in its most developed and accurate form; the following introduces the basic concepts of classical mechanics. For simplicity, it models real-world objects as point particles; the motion of a point particle is characterized by a small number of parameters: its position and the forces applied to it. Each of these parameters is discussed in turn. In reality, the kind of objects that classical mechanics can describe always have a non-zero size.
Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the additional degrees of freedom, e.g. a baseball can spin while it is moving. However, the results for point particles can be used to study such objects by treating them as composite objects, made of a large number of collectively acting point particles; the center of mass of a composite object behaves like a point particle. Classical mechanics uses common-sense notions of how matter and forces interact, it assumes that matter and energy have definite, knowable attributes such as location in space and speed. Non-relativistic mechanics assumes that forces act instantaneously; the position of a point particle is defined in relation to a coordinate system centered on an arbitrary fixed reference point in space called the origin O. A simple coordinate system might describe the position of a particle P with a vector notated by an arrow labeled r that points from the origin O to point P. In general, the point particle does not need to be stationary relative to O.
In cases where P is moving relative to O, r is defined as a function of time. In pre-Einstein relativity, time is considered an absolute, i.e. the time interval, observed to elapse between any given pair of events is the same for all observers. In addition to relying on absolute time, classical mechanics assumes Euclidean geometry for the structure of space; the velocity, or the rate of change of position with time, is defined as the derivative of the position with respect to time: v = d r d t. In classical mechanics, velocities are directly subtractive. For example, if one car travels east at 60 km/h and passes another car traveling in the same direction at 50 km/h, the slower car perceives the faster car as traveling east at 60 − 50 = 10 km/h. However, from the perspective of the faster car, the slower car is moving 10 km/h to the west denoted as -10 km/h where the sign implies opposite direction. Velocities are directly additive as vector quantities. Mathematically, if the velocity of the first object in the previous discussion is denoted by the vector u = ud and the velocity of the second object by the vector v = ve, where u is the speed of the first object, v is the speed of the second object, d and e are unit vectors in the directions of motion of each object then the velocity of the first object as seen by the second object is u ′ = u − v. Similarly, the first object sees the velocity of the second object as v ′ = v − u.
When both objects are moving in the same direction, this equation can be simplified to u ′ = d. Or, by ignoring direction, the difference can be given in terms of speed only: u ′ = u − v; the acceleration, or rate of change of velocity, is th
In classical mechanics, impulse is the integral of a force, F, over the time interval, t, for which it acts. Since force is a vector quantity, impulse is a vector in the same direction. Impulse applied to an object produces an equivalent vector change in its linear momentum in the same direction; the SI unit of impulse is the newton second, the dimensionally equivalent unit of momentum is the kilogram meter per second. The corresponding English engineering units are the slug-foot per second. A resultant force causes acceleration and a change in the velocity of the body for as long as it acts. A resultant force applied over a longer time therefore produces a bigger change in linear momentum than the same force applied briefly: the change in momentum is equal to the product of the average force and duration. Conversely, a small force applied for a long time produces the same change in momentum—the same impulse—as a larger force applied briefly. J = F average The impulse is the integral of the resultant force with respect to time: J = ∫ F d t Impulse J produced from time t1 to t2 is defined to be J = ∫ t 1 t 2 F d t where F is the resultant force applied from t1 to t2.
From Newton's second law, force is related to momentum p by F = d p d t Therefore, J = ∫ t 1 t 2 d p d t d t = ∫ p 1 p 2 d p = p 2 − p 1 = Δ p where Δp is the change in linear momentum from time t1 to t2. This is called the impulse-momentum theorem; as a result, an impulse may be regarded as the change in momentum of an object to which a resultant force is applied. The impulse may be expressed in a simpler form when the mass is constant: J = ∫ t 1 t 2 F d t = Δ p = m v 2 − m v 1 where F is the resultant force applied, t1 and t2 are times when the impulse begins and ends m is the mass of the object, v2 is the final velocity of the object at the end of the time interval, v1 is the initial velocity of the object when the time interval begins. Impulse has the same dimensions as momentum. In the International System of Units, these are kg⋅m/s = N⋅s. In English engineering units, they are slug⋅ft/s = lbf⋅s; the term "impulse" is used to refer to a fast-acting force or impact. This type of impulse is idealized so that the change in momentum produced by the force happens with no change in time.
This sort of change is a step change, is not physically possible. However, this is a useful model for computing the effects of ideal collisions. Additionally, in rocketry, the term "total impulse" is used and is considered synonymous with the term "impulse"; the application of Newton's second law for variable mass allows impulse and momentum to be used as analysis tools for jet- or rocket-propelled vehicles. In the case of rockets, the impulse imparted can be normalized by unit of propellant expended, to create a performance parameter, specific impulse; this fact can be used to derive the Tsiolkovsky rocket equation, which relates the vehicle's propulsive change in velocity to the engine's specific impulse and the vehicle's propellant-mass ratio. Wave–particle duality defines the impulse of a wave collision; the preservation of momentum in the collision is called phase matching. Applications include: Compton effect Nonlinear optics Acousto-optic modulator Electron phonon scattering Serway, Raymond A..
Physics for Scientists and Engineers. Brooks/Cole. ISBN 0-534-40842-7. Tipler, Paul. Physics for Scientists and Engineers: Mechanics and Waves, Thermodynamics. W. H. Freeman. ISBN 0-7167-0809-4. Dynamics