An example of a mechanical system: The only force acting on a satellite orbiting the Earth is its own weight; its mechanical energy is therefore conserved. The satellite's acceleration is represented by the green vector and its velocity is represented by the red vector. The potential energy of the satellite, and its kinetic energy, may vary with time but their sum remains constant.
Inphysical science,mechanical energy is the sum of macroscopicpotential andkinetic energies. The principle of conservation of mechanical energy states that if anisolatedsystem or a closed system is subject only toconservative forces, then the mechanical energy is constant.[1] If an object moves in the opposite direction of a conservative net force, the potential energy will increase; and if thespeed (not thevelocity) of the object changes, the kinetic energy of the object also changes. In all real systems, however,nonconservative forces, such asfrictional forces, will be present, but if they are of negligiblemagnitude, the mechanical energy changes little and its conservation is a useful approximation. Inelastic collisions, the kinetic energy is conserved, but ininelastic collisions some mechanical energy may be converted intothermal energy. The equivalence between lost mechanical energy and an increase intemperature was discovered byJames Prescott Joule.
Energy is ascalar quantity, and the mechanical energy of a system is the sum of the potential energy (which is measured by the position of the parts of the system) and the kinetic energy (which is also called the energy of motion):[2][3]
The potential energy,U, depends on the position of an object subjected to gravity or some otherconservative force. The gravitational potential energy of an object is equal to the weightW of the object multiplied by the heighth of the object's center of gravity relative to an arbitrary datum:
Potential energy is the energy stored in an object due to its position relative to a conservative force field, such as gravity or a spring. It increases when work is done against the force—meaning when the object is moved in the direction opposite to that of the force.[nb 1][2] IfF represents the conservative force andx the position, the potential energy of the force between the two positionsx1 andx2 is defined as the negative integral ofF fromx1 tox2:[5]
The kinetic energy,K, depends on the speed of an object and is the ability of a moving object to do work on other objects when it collides with them.[nb 2][9] It is defined as one half the product of the object's mass with the square of its speed, and the total kinetic energy of a system of objects is the sum of the kinetic energies of the respective objects:[2][10]
The principle of conservation of mechanical energy states that if a body or system is subjected only toconservative forces, the mechanical energy of that body or system remains constant.[11] The difference between a conservative and anon-conservative force is that when a conservative force moves an object from one point to another, the work done by the conservative force is independent of the path. On the contrary, when a non-conservative force acts upon an object, the work done by the non-conservative force is dependent of the path.[12][13]
MIT professorWalter Lewin demonstrating conservation of mechanical energy
According to the principle of conservation of mechanical energy, the mechanical energy of anisolated system remains constant in time, as long as the system is free offriction and other non-conservative forces. In any real situation, frictional forces and other non-conservative forces are present, but in many cases their effects on the system are so small that the principle of conservation of mechanical energy can be used as a fairapproximation. Though energy cannot be created or destroyed, it can beconverted to another form of energy.[2][14]
A swinging pendulum with the velocity vector (green) and acceleration vector (blue). The magnitude of the velocity vector, the speed, of the pendulum is greatest in the vertical position and the pendulum is farthest from Earth in its extreme positions.
In amechanical system like a swingingpendulum subjected to the conservativegravitational force where frictional forces like air drag and friction at the pivot are negligible, energy passes back and forth between kinetic and potential energy but never leaves the system. The pendulum reaches greatest kinetic energy and least potential energy when in the vertical position, because it will have the greatest speed and be nearest the Earth at this point. On the other hand, it will have its least kinetic energy and greatest potential energy at the extreme positions of its swing, because it has zero speed and is farthest from Earth at these points. However, when taking the frictional forces into account, the system loses mechanical energy with each swing because of the negative work done on the pendulum by these non-conservative forces.[3]
That the loss of mechanical energy in a system always resulted in an increase of the system's temperature has been known for a long time, but it was the amateur physicistJames Prescott Joule who first experimentally demonstrated how a certain amount of work done against friction resulted in a definite quantity ofheat which should be conceived as the random motions of the particles that comprise matter.[15] This equivalence between mechanical energy and heat is especially important when considering colliding objects. In anelastic collision, mechanical energy is conserved – the sum of the mechanical energies of the colliding objects is the same before and after the collision. After aninelastic collision, however, the mechanical energy of the system will have changed. Usually, the mechanical energy before the collision is greater than the mechanical energy after the collision. In inelastic collisions, some of the mechanical energy of the colliding objects is transformed into kinetic energy of the constituent particles. This increase in kinetic energy of the constituent particles is perceived as an increase in temperature. The collision can be described by saying some of the mechanical energy of the colliding objects has been converted into an equal amount of heat. Thus, the total energy of the system remains unchanged though the mechanical energy of the system has reduced.[2][16]
plot of kinetic energy, gravitational potential energy, and mechanical energy versus distance away from centre of earth, r at R= Re, R= 2*Re, R=3*Re and lastly R = geostationary radius
A satellite of mass at a distance from the centre of Earth possesses both kinetic energy,, (by virtue of its motion) and gravitational potential energy,, (by virtue of its position within the Earth's gravitational field; Earth's mass is).Hence, mechanical energy of the satellite-Earth system is given by
If the satellite is in circular orbit, the energy conservation equation can be further simplified intosince in circular motion, Newton's 2nd Law of motion can be taken to be
^When measuring mechanical energy, an object is considered as a whole, as it is stated byIsaac Newton in hisPrincipia: "The motion of a whole is the same as the sum of the motions of the parts; that is, the change in position of its parts from their places, and thus the place of a whole is the same as the sum of the places of the parts and therefore is internal and in the whole body."[4]
^In physics,speed is a scalar quantity andvelocity is avector. Velocity is speed with a direction and can therefore change without changing the speed of the object since speed is the numerical magnitude of a velocity.[6][7][8]
^Resnick, Robert and Halliday, David (1966),Physics, Section 8-3 (Vol I and II, Combined edition), Wiley International Edition, Library of Congress Catalog Card No. 66-11527
^E. Roller, Duane; Leo Nedelsky (2008)."Conservation of energy".AccessScience. McGraw-Hill Companies. Retrieved2011-08-26.
^"James Prescott Joule".Scientists: Their Lives and Works. Gale. 2006. as cited on"Student Resources in Context". Gale. Retrieved2011-08-28.
^Schmidt, Paul W. (2008)."Collision (physics)".AccessScience. McGraw-Hill Companies. Retrieved2011-09-03.
^Kopicki, Ronald J. (2003). "Electrification, Household". In Kutler, Stanley I. (ed.).Dictionary of American History. Vol. 3 (3rd ed.). New York: Charles Scribner's Sons. pp. 179–183. as cited on"Student Resources in Context". Gale. Retrieved2011-09-07.
^Lerner, K. Lee; Lerner, Brenda Wilmoth, eds. (2008). "Electric motor".The Gale Encyclopedia of Science (4th ed.). Detroit: Gale. as cited on"Student Resources in Context". Gale. Retrieved2011-09-07.
^"Electric motor".U*X*L Encyclopedia of Science. U*X*L. 2007. as cited on"Student Resources in Context". Gale. Retrieved2011-09-07.
^"Generator".U*X*L Encyclopedia of Science. U*X*L. 2007-07-16. as cited on"Student Resources in Context". Gale. Retrieved2011-10-09.
^Lerner, K. Lee; Lerner, Brenda Wilmoth, eds. (2008). "Internal combustion engine".The Gale Encyclopedia of Science (4th ed.). Detroit: Gale. as cited on"Student Resources in Context". Gale. Retrieved2011-10-09.
^"Steam engine".U*X*L Encyclopedia of Science. U*X*L. 2007-07-16. as cited on"Student Resources in Context". Gale. Retrieved2011-10-09.
^Lerner, K. Lee; Lerner, Brenda Wilmoth, eds. (2008). "Turbine".The Gale Encyclopedia of Science (4th ed.). Detroit: Gale. as cited on"Student Resources in Context". Gale. Retrieved2011-10-09.
^Atkins, Peter W. (2008)."Chemical energy".AccessScience. McGraw-Hill Companies. Archived fromthe original on 2013-07-19. Retrieved2011-10-17.
^Duckworth, Henry E.; Wilkinson, D. H. (2008)."Nuclear binding energy".AccessScience. McGraw-Hill Companies. Archived fromthe original on 2013-07-19. Retrieved2011-10-17.
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