# Work, Energy and Power

### Introduction

The notions of power, energy and work are based on the common representation of these concepts. However, in everyday life people use these concepts in more broad and general meaning than they have in physics. For example, work in everyday life is understood as a physical activity performed by an individual; power is the speed of activity performance, and energy is the person’s capability to perform this activity. Similarly, in physics, there is also a direct link between work, energy and power; they intersect, complete and define each other. However, in physics these notions are more complicated and have a mathematical representation. The concepts of work, energy and power are fundamental physical quantities, without which it is impossible to get a deeper knowledge of physics and any other related science.

## Energy

Energy is a scalar (i.e. having a particular value) quantity identifying what kind of work a body can perform. Mechanical energy is equal to mechanical work that the body can perform under certain conditions. Mechanical work is a measure of energy change in a variety of processes. Therefore, energy and work are expressed in the same units (SI - in Joules).

In a more general sense, energy is a single measure for different forms of motion as well as a measure of transition movement of a matter from one state to another. Characteristics for specific forms of motion use the concept of corresponding energy types: mechanical, internal, electromagnetic, etc. Energy of the material body systems characterizes the system in terms of the possible quantitative and qualitative movement transformations. These transformations are due to both the interaction of bodies among themselves and with the external systems of bodies.

In mechanics, it is decided to distinguish between two types of mechanical energy: the energy produced by the motion of bodies, or kinetic energy, and energy produced by the interaction of bodies, or potential energy (Halliday, Resnick, and Walker 140). Kinetic energy determines the state of the body and does not depend on the manner in which the body comes to this state. This value of energy is determined by the weight (mass) of the body (m) and its movement speed (v) in the particular frame of reference, i.e. is a function of its motion.

Kinetic energy is relative since the velocity of the body depends on the choice of the reference system in which the motion is considered. In different inertial frames of reference moving relative to each other, the speed of the body and thus its kinetic energy will vary. Kinetic energy change is possible only by doing work. Here, kinetic energy is equal to the algebraic sum of all external forces acting on the body during this period.

Potential energy is energy generated by the mutual arrangement of interacting bodies (or parts of the body). This type of energy is the product of the body’s mass (m), free fall acceleration (g=9.8m/s2) and height (h) at which the body is placed .

The concept of potential energy is introduced only for such forces whose work does not depend on the shape of the trajectory and depends only on the location of the final and initial points of the trajectory.

## The Law of Energy Conservation

The abovementioned types of energy are connected by the most widely used and fundamental law of physics, the law of energy conservation. The idea of the law belongs to Lomonosov (1711-1765), who set out the law of matter and motion conservation, and the quantitative formulation of the law was given by the German physician Mayer (1814-1878) and the German naturalist Helmholtz (1821-1894). The law states that the total mechanical energy of the system (W) is the sum of kinetic (Wk) and potential energies (Wp) of all the bodies of the system.

The law of conservation and transformation of energy (the full name of the mentioned law) is a fundamental law of nature, which is fair for both macroscopic bodies and micro particle systems. Strictly speaking, all real macroscopic systems in nature are dissipative, i.e. undergo the influence of external forces, such as friction. In dissipative systems, the total mechanical energy is gradually reduced due to conversion to other (non-mechanical) forms of energy, such as internal energy. The internal energy of folding is the kinetic energy of the invisible thermal atoms and molecules motion and the potential energy of their interaction. Thermal motion of atoms and molecules is perceived by human senses in the form of heat.

A closed system of bodies is an idealized model for simplified consideration of many phenomena and processes. Therefore, in actual cases, the law of conservation of mechanical energy is not met. However, the “disappearance” of mechanical energy always corresponds to the production of an equivalent amount of energy of another kind. Thus, the energy never disappears and reappears, it only transforms from one form to another or passes from one body to another. The total amount of energy remains constant. This is the physical nature of the law of conservation and transformation of energy.

## Work

The forces acting on the body cause the change in the mechanical movement of this body. In order to characterize energy exchange between interacting bodies quantitatively, the concept of work force is introduced. Work is a measure of energy change. If the body moves forward and a constant force F act upon it, which makes an angle α with the direction displacement S, the elementary work A of this force is equal to the scalar product of these vectors multiplied by the cosine of the angle between them.

The dimension of work: [A] = [F][S] = 1 m·1N= 1 Joule = 1 2 kg×m/s2. One joule is the work of 1N force at the body’s motion at the displacement on the distance of 1 m in the direction of the force.

In the ordinary sense, any effort, in particular muscular tension, is always accompanied by the performance of work. For example, to keep a heavy load while standing still or moving it horizontally, a porter spends a lot of effort, i.e. performs work. However, work as a mechanical value in these case is zero, and the energy of the load does not change.

Taking into account that work can be both positive and negative, it is an algebraic value. If force is applied to the body and performs work, the speed of the body increases. Indeed, in this case, force and acceleration are directed along the velocity, increasing it. If force performs a negative work, acceleration is directed against the speed, and velocity of the body decreases. Therefore, in general, work can be varied as modulus by both positive and negative direction.

## Power

Power is a quantity that characterizes the speed of execution of work and is equal to the ratio of work A performed at some time interval.

[P] = [A]/[t]=J/s= Watt. 1 Watt is the power at which 1 Joule of work is done in 1 second.

When people talk about power (or work), it is necessary to be clear about which particular work (power) of what force or forces is discussed. From the definition of work, a mechanical work is performed in case a single body acts on another body and causes its movement and the direction of force is not perpendicular to the direction of the movement. Work can be made by any body. Therefore, the body produces power by performing any work. For example, a compressed or stretched spring acts on the elastic attached to its body and moves it, at the same time performing mechanical work.

## Hooke’s Law

Since at deformation a spring also performs work, has energy and power, its motion also can be described in terms of these concepts. The force F acting upon the spring and performing work is called strain energy and is represented by the product of elasticity coefficient k, describing the body’s rigidity and linear displacement of the body x (stretching or compression).

The minus sign in the formula indicates that the spring counteracts the deformation: seeks to shorten at stretching and to lengthen at compression. Knowing the strain energy, one can define the work and power of the spring. In addition, the Hooke’s law started a new science, called the elasticity theory.

### Conclusion

Therefore, the mentioned three quantities and the described laws are the fundamental concept in mechanics. These notions describe the simplest and typical motion of bodies and particles, being true both for macro- and micro particles systems. A direct link between work, energy and power allows one to understand the principles of these concepts’ work as well as draw a conclusion about the importance, derivation and action of them upon each other. Knowing the essence of work, energy and power is the basis for the further study of mechanics and mechanical motion.