# Mechanical energy

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In physical sciences, mechanical energy is the sum of potential energy and kinetic energy. It is the macroscopic energy associated with a system. The principle of conservation of mechanical energy states that in an isolated system that is only subject to conservative forces, the mechanical energy is constant. If an object moves in the opposite direction of a conservative net force, the potential energy will increase; and if the speed (not the velocity) of the object changes, the kinetic energy of the object also changes. In all real systems, however, nonconservative forces, such as frictional forces, will be present, but if they are of negligible magnitude, the mechanical energy changes little and its conservation is a useful approximation. In elastic collisions, the mechanical energy is conserved, but in inelastic collisions some mechanical energy is converted into thermal energy. The equivalence between lost mechanical energy (dissipation) and an increase in temperature was discovered by James Prescott Joule.

## Contents

Many devices are used to convert mechanical energy to or from other forms of energy, e.g. an electric motor converts electrical energy to mechanical energy, an electric generator converts mechanical energy into electrical energy and a heat engine converts heat energy to mechanical energy.

## General

Energy is a scalar 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): [1] [2]

${\displaystyle E_{\mathrm {mechanical} }=U+K\,}$

The potential energy, U, depends on the position of an object subjected to a conservative force. It is defined as the object's ability to do work and is increased as the object is moved in the opposite direction of the direction of the force. [nb 1] [1] If F represents the conservative force and x the position, the potential energy of the force between the two positions x1 and x2 is defined as the negative integral of F from x1 to x2: [4]

${\displaystyle U=-\int _{x_{1}}^{x_{2}}{\vec {F}}\cdot d{\vec {x}}}$

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] [8] 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: [1] [9]

${\displaystyle K={1 \over 2}mv^{2}}$

The principle of conservation of mechanical energy states that if a body or system is subjected only to conservative forces, the mechanical energy of that body or system remains constant. [10] The difference between a conservative and a non-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. [11] [12]

## Conservation of mechanical energy

According to the principle of conservation of mechanical energy, the mechanical energy of an isolated system remains constant in time, as long as the system is free of friction 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 fair approximation. Though energy cannot be created or destroyed in an isolated system, it can be converted to another form of energy. [1] [13]

### Swinging pendulum

In a mechanical system like a swinging pendulum subjected to the conservative gravitational 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. [2]

### Irreversibilities

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 physicist James Prescott Joule who first experimentally demonstrated how a certain amount of work done against friction resulted in a definite quantity of heat which should be conceived as the random motions of the particles that comprise matter. [14] This equivalence between mechanical energy and heat is especially important when considering colliding objects. In an elastic collision, mechanical energy is conserved – the sum of the mechanical energies of the colliding objects is the same before and after the collision. After an inelastic 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. [1] [15]

### Satellite

A satellite of mass ${\displaystyle m}$ at a distance ${\displaystyle r}$ from the centre of Earth possesses both kinetic energy, ${\displaystyle K}$, (by virtue of its motion) and gravitational potential energy, ${\displaystyle U}$, (by virtue of its position within the Earth's gravitational field; Earth's mass is ${\displaystyle M}$). Hence, mechanical energy ${\displaystyle E_{\mathrm {mechanical} }}$ of the satellite-Earth system is given by

${\displaystyle E_{\mathrm {mechanical} }=U+K}$
${\displaystyle E_{\mathrm {mechanical} }=-G{\frac {Mm}{r}}\ +{\frac {1}{2}}\ mv^{2}}$

If the satellite is in circular orbit, the energy conservation equation can be further simplified into

${\displaystyle E_{\mathrm {mechanical} }=-G{\frac {Mm}{2r}}\ }$

since in circular motion, Newton's 2nd Law of motion can be taken to be

${\displaystyle G{\frac {Mm}{r^{2}}}\ ={\frac {mv^{2}}{r}}\ }$

## Conversion

Today, many technological devices convert mechanical energy into other forms of energy or vice versa. These devices can be placed in these categories:

## Distinction from other types

The classification of energy into different types often follows the boundaries of the fields of study in the natural sciences.

## Related Research Articles

In physics, energy is the quantitative property that must be transferred to an object in order to perform work on, or to heat, the object. Energy is a conserved quantity; the law of conservation of energy states that energy can be converted in form, but not created or destroyed. The SI unit of energy is the joule, which is the energy transferred to an object by the work of moving it a distance of 1 metre against a force of 1 newton.

In physics, a force is any interaction that, when unopposed, will change the motion of an object. A force can cause an object with mass to change its velocity, i.e., to accelerate. Force can also be described intuitively as a push or a pull. A force has both magnitude and direction, making it a vector quantity. It is measured in the SI unit of newtons and represented by the symbol F.

Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:

In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x:

In physics, the kinetic energy (KE) 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 Newtonian mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If m is an object's mass and v is its velocity, then the object's momentum is:

In SI units, momentum is measured in kilogram meters per second (kg⋅m/s).

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.

In physics, escape velocity is the minimum speed needed for a free, non-propelled object to escape from the gravitational influence of a massive body, that is, to achieve an infinite distance from it. Escape velocity is a function of the mass of the object and distance to the center of mass of the object.

A collision is the event in which two or more bodies exert forces on each other in about a relatively short time. Although the most common use of the word collision refers to incidents in which two or more objects collide with great force, the scientific use of the term implies nothing about the magnitude of the force.

A conservative force is a force with the property that the total work done in moving a particle between two points is independent of the taken path. Equivalently, if a particle travels in a closed loop, the total work done by a conservative force is zero.

A brake is a mechanical device that inhibits motion by absorbing energy from a moving system. It is used for slowing or stopping a moving vehicle, wheel, axle, or to prevent its motion, most often accomplished by means of friction.

An inelastic collision, in contrast to an elastic collision, is a collision in which kinetic energy is not conserved due to the action of internal friction.

In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be conserved over time. This law, first proposed and tested by Émilie du Châtelet, means that energy can neither be created nor destroyed; rather, it can only be transformed or transferred from one form to another. For instance, chemical energy is converted to kinetic energy when a stick of dynamite explodes. If one adds up all forms of energy that were released in the explosion, such as the kinetic energy and potential energy of the pieces, as well as heat and sound, one will get the exact decrease of chemical energy in the combustion of the dynamite. Classically, conservation of energy was distinct from conservation of mass; however, special relativity showed that mass is related to energy and vice versa by E = mc2, and science now takes the view that mass–energy as a whole is conserved. Theoretically, this implies that any object with mass can itself be converted to pure energy, and vice versa, though this is believed to be possible only under the most extreme of physical conditions, such as likely existed in the universe very shortly after the Big Bang.

In thermodynamics, the internal energy of a system is the energy contained within the system. It is the energy necessary to create or prepare the system in any given state, but does not include the kinetic energy of motion of the system as a whole, nor the potential energy of the system as a whole due to external force fields which includes the energy of displacement of the system's surroundings. It keeps account of the gains and losses of energy of the system that are due to changes in its internal state.

In physics, mass–energy equivalence is the principle that anything having mass has an equivalent amount of energy and vice versa, with these fundamental quantities directly relating to one another by Albert Einstein's famous formula:

In physics, relativistic mechanics refers to mechanics compatible with special relativity (SR) and general relativity (GR). It provides a non-quantum mechanical description of a system of particles, or of a fluid, in cases where the velocities of moving objects are comparable to the speed of light c. As a result, classical mechanics is extended correctly to particles traveling at high velocities and energies, and provides a consistent inclusion of electromagnetism with the mechanics of particles. This was not possible in Galilean relativity, where it would be permitted for particles and light to travel at any speed, including faster than light. The foundations of relativistic mechanics are the postulates of special relativity and general relativity. The unification of SR with quantum mechanics is relativistic quantum mechanics, while attempts for that of GR is quantum gravity, an unsolved problem in physics.

The coefficient of restitution (COR), also denoted by (e), is the ratio of the final to initial relative velocity between two objects after they collide. It normally ranges from 0 to 1 where 1 would be a perfectly elastic collision. A perfectly inelastic collision has a coefficient of 0, but a 0 value does not have to be perfectly inelastic. It is measured in the Leeb rebound hardness test, expressed as 1000 times the COR, but it is only a valid COR for the test, not as a universal COR for the material being tested.

Coulomb damping is a type of constant mechanical damping in which energy is absorbed via sliding friction. The friction generated by the relative motion of the two surfaces that press against each other is a source of energy dissipation. In general, damping is the dissipation of energy from a vibrating system where the kinetic energy is converted into heat by the friction. Coulomb damping is a common damping mechanism that occurs in machinery.

Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars and galaxies.

In the context of classical mechanics simulations and physics engines employed within video games, collision response deals with models and algorithms for simulating the changes in the motion of two solid bodies following collision and other forms of contact.

## References

Notes

1. It is important to note that when measuring mechanical energy, an object is considered as a whole, as it is stated by Isaac Newton in his Principia : "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." [3]
2. In physics, speed is a scalar quantity and velocity is a vector. In other words, 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. [5] [6] [7]

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