Mechanical energy

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An example of a mechanical system: A satellite is orbiting the Earth influenced only by the conservative gravitational force; 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. If the satellite's orbit is an ellipse the potential energy of the satellite, and its kinetic energy, both vary with time but their sum remains constant. Orbital motion.gif
An example of a mechanical system: A satellite is orbiting the Earth influenced only by the conservative gravitational force; 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. If the satellite's orbit is an ellipse the potential energy of the satellite, and its kinetic energy, both vary with time but their sum remains constant.

In physical sciences, mechanical energy is the sum of potential energy and kinetic energy. The principle of conservation of mechanical energy states that if an isolated system is subject only to conservative forces, then 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 kinetic energy is conserved, but in inelastic collisions some mechanical energy may be converted into thermal energy. The equivalence between lost mechanical energy 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 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]

The potential energy, U, depends on the position of an object subjected to gravity or some other conservative force. The gravitational potential energy of an object is equal to the weight W of the object multiplied by the height h of the object's center of gravity relative to an arbitrary datum:

The potential energy of an object can be 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]

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]

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

MIT professor Walter Lewin demonstrating 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, it can be converted to another form of energy. [1] [13]

Swinging pendulum

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. Pendulum animation.gif
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 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

plot of kinetic energy
K
{\displaystyle K}
, gravitational potential energy,
U
{\displaystyle U}
and mechanical energy
E
mechanical
{\displaystyle E_{\text{mechanical}}}
versus distance away from centre of earth, r at R= Re, R= 2*Re, R=3*Re and lastly R = geostationary radius R = geo Re 2012-10-08 1809.png
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 into

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

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

<span class="mw-page-title-main">Energy</span> Physical quantity

In physics, energy is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat and light. 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 unit of measurement for energy in the International System of Units (SI) is the joule (J).

<span class="mw-page-title-main">Force</span> Influence that can change motion of an object

In physics, a force is an influence that can cause an object to change its velocity, i.e., to accelerate, meaning a change in speed or direction, unless counterbalanced by other forces. The concept of force makes the everyday notion of pushing or pulling mathematically precise. Because the magnitude and direction of a force are both important, force is a vector quantity. The SI unit of force is the newton (N), and force is often represented by the symbol F.

<span class="mw-page-title-main">Friction</span> Force resisting sliding motion

Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. Types of friction include dry, fluid, lubricated, skin, and internal.

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:

<span class="mw-page-title-main">Kinetic energy</span> Energy of a moving physical body

In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion.

<span class="mw-page-title-main">Momentum</span> Property of a mass in motion

In Newtonian mechanics, 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 p is:

<span class="mw-page-title-main">Potential energy</span> Energy held by an object because of its position relative to other objects

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. The term potential energy was introduced by the 19th-century Scottish engineer and physicist William Rankine, although it has links to the ancient Greek philosopher Aristotle's concept of potentiality.

<span class="mw-page-title-main">Escape velocity</span> Concept in celestial mechanics

In celestial mechanics, escape velocity or escape speed is the minimum speed needed for an object to escape from contact with or orbit of a primary body, assuming:

<span class="mw-page-title-main">Collision</span> Instance of two or more bodies physically contacting each other within a short period of time

In physics, a collision is any event in which two or more bodies exert forces on each other in 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.

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

<span class="mw-page-title-main">Inelastic collision</span> Collision in which energy is lost to heat

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.

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. In the case of a closed system the principle says that the total amount of energy within the system can only be changed through energy entering or leaving the system. 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.

<span class="mw-page-title-main">Internal energy</span> Energy contained within a system

The internal energy of a thermodynamic system is the energy contained within it, measured as the quantity of energy necessary to bring the system from its standard internal state to its present internal state of interest, accounting for the gains and losses of energy due to changes in its internal state, including such quantities as magnetization. It excludes the kinetic energy of motion of the system as a whole and the potential energy of position of the system as a whole, with respect to its surroundings and external force fields. It includes the thermal energy, i.e., the constituent particles' kinetic energies of motion relative to the motion of the system as a whole. The internal energy of an isolated system cannot change, as expressed in the law of conservation of energy, a foundation of the first law of thermodynamics.

<span class="mw-page-title-main">Thermal energy</span> Energy that is measured by temperature

The term "thermal energy" is used loosely in various contexts in physics and engineering, generally related to the kinetic energy of vibrating and colliding atoms in a substance. It can refer to several different physical concepts. These include the internal energy or enthalpy of a body of matter and radiation; heat, defined as a type of energy transfer ; and the characteristic energy of a degree of freedom, , in a system that is described in terms of its microscopic particulate constituents.

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.

<span class="mw-page-title-main">Coefficient of restitution</span> Ratio characterising inelastic collisions

The coefficient of restitution, is the ratio of the relative velocity of separation after collision to the relative velocity of approach before collision. It can aIso be defined as the square root of the ratio of the final kinetic energy to the initial kinetic energy. 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 the system's kinetic energy is absorbed via sliding friction. Coulomb damping is a common damping mechanism that occurs in machinery.

<span class="mw-page-title-main">Classical mechanics</span> Description of large objects physics

Classical mechanics is a physical theory describing the motion of objects such as projectiles, parts of machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics involved substantial change in the methods and philosophy of physics. The qualifier classical distinguishes this type of mechanics from physics developed after the revolutions in physics of the early 20th century, all of which revealed limitations in classical mechanics.

This glossary of engineering terms is a list of definitions about the major concepts of engineering. Please see the bottom of the page for glossaries of specific fields of engineering.

<span class="mw-page-title-main">Bouncing ball</span> Physics of bouncing balls

The physics of a bouncing ball concerns the physical behaviour of bouncing balls, particularly its motion before, during, and after impact against the surface of another body. Several aspects of a bouncing ball's behaviour serve as an introduction to mechanics in high school or undergraduate level physics courses. However, the exact modelling of the behaviour is complex and of interest in sports engineering.

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. 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]

Citations

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  3. Newton 1999, p. 409
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  5. Brodie et al. 1998, pp. 129–131
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  8. Brodie et al. 1998 , p. 101
  9. Jain 2009 , p. 9
  10. Jain 2009 , p. 12
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