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-
In physics, power is the rate of doing work or of transferring heat, i.e. the amount of energy transferred or converted per unit time. Having no direction, it is a scalar quantity. In the International System of Units, the unit of power is the joule per second, known as the watt in honour of James Watt, the eighteenth-century developer of the condenser steam engine. Another common and traditional measure is horsepower. Being the rate of work, the equation for power can be written: power = work time As a physical concept, power requires both a change in the physical system and a specified time in which the change occurs; this is distinct from the concept of work, only measured in terms of a net change in the state of the physical system. The same amount of work is done when carrying a load up a flight of stairs whether the person carrying it walks or runs, but more power is needed for running because the work is done in a shorter amount of time; the output power of an electric motor is the product of the torque that the motor generates and the angular velocity of its output shaft.
The power involved in moving a vehicle is the product of the traction force of the wheels and the velocity of the vehicle. The rate at which a light bulb converts electrical energy into light and heat is measured in watts—the higher the wattage, the more power, or equivalently the more electrical energy is used per unit time; the dimension of power is energy divided by time. The SI unit of power is the watt, equal to one joule per second. Other units of power include ergs per second, metric horsepower, foot-pounds per minute. One horsepower is equivalent to 33,000 foot-pounds per minute, or the power required to lift 550 pounds by one foot in one second, is equivalent to about 746 watts. Other units include a logarithmic measure relative to a reference of 1 milliwatt. Power, as a function of time, is the rate at which work is done, so can be expressed by this equation: P = d W d t where P is power, W is work, t is time; because work is a force F applied over a distance x, W = F ⋅ x for a constant force, power can be rewritten as: P = d W d t = d d t = F ⋅ d x d t = F ⋅ v In fact, this is valid for any force, as a consequence of applying the fundamental theorem of calculus.
As a simple example, burning one kilogram of coal releases much more energy than does detonating a kilogram of TNT, but because the TNT reaction releases energy much more it delivers far more power than the coal. If ΔW is the amount of work performed during a period of time of duration Δt, the average power Pavg over that period is given by the formula P a v g = Δ W Δ t, it is the average amount of energy converted per unit of time. The average power is simply called "power" when the context makes it clear; the instantaneous power is the limiting value of the average power as the time interval Δt approaches zero. P = lim Δ t → 0 P a v g = lim Δ t → 0 Δ W Δ t = d W d t. In the case of constant power P, the amount of work performed during a period of duration t is given by: W = P t. In the context of energy conversion, it is more customary to use the symbol E rather than W. Power in mechanical systems is the combination of forces and movement. In particular, power is the product of a force on an object and the object's velocity, or the product of a torque on a shaft and the shaft's angular velocity.
Mechanical power is described as the time derivative of work. In mechanics, the work done by a force F on an object that travels along a curve C is given by the line integral: W C = ∫ C F ⋅ v d t = ∫ C F ⋅ d x, where x defines the path C and v is the velocity along this path. If the force F is derivable from a potential applying the gradi
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(
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
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, ∂
Rigid body dynamics
Rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces. The assumption that the bodies are rigid, which means that they do not deform under the action of applied forces, simplifies the analysis by reducing the parameters that describe the configuration of the system to the translation and rotation of reference frames attached to each body; this excludes bodies that display fluid elastic, plastic behavior. The dynamics of a rigid body system is described by the laws of kinematics and by the application of Newton's second law or their derivative form Lagrangian mechanics; the solution of these equations of motion provides a description of the position, the motion and the acceleration of the individual components of the system and overall the system itself, as a function of time. The formulation and solution of rigid body dynamics is an important tool in the computer simulation of mechanical systems. If a system of particles moves parallel to a fixed plane, the system is said to be constrained to planar movement.
In this case, Newton's laws for a rigid system of N particles, Pi, i=1... N, simplify. Determine the resultant force and torque at a reference point R, to obtain F = ∑ i = 1 N m i A i, T = ∑ i = 1 N ×, where ri denotes the planar trajectory of each particle; the kinematics of a rigid body yields the formula for the acceleration of the particle Pi in terms of the position R and acceleration A of the reference particle as well as the angular velocity vector ω and angular acceleration vector α of the rigid system of particles as, A i = α × + ω × + A. For systems that are constrained to planar movement, the angular velocity and angular acceleration vectors are directed along k perpendicular to the plane of movement, which simplifies this acceleration equation. In this case, the acceleration vectors can be simplified by introducing the unit vectors ei from the reference point R to a point ri and the unit vectors t i = k × e i, so A i = α − ω 2 + A; this yields the resultant force on the system as F = α ∑ i = 1 N m i − ω 2 ∑ i = 1 N m i + A, torque as T = ∑ i = 1 N × = α k → + × A, where e i × e i = 0 and e i × t i = k is the unit