Energy of Point Object Systems

Energy refers to a system’s ability to do work or to make something happen.7 This helps us understand that different forms of energy have the capacity to perform different actions. For example, mechanical energy can cause objects to move or accelerate. An ice cube sitting on the kitchen counter at room temperature will absorb thermal energy through heat transfer and eventually melt into water, undergoing a phase transformation from solid to liquid. Kinetic energy is the energy of motion.8 Objects that have mass and that are moving with some speed will have an associated amount of kinetic energy, calculated as follows:

K = ½ mv2

where K is kinetic energy, m is the mass in kilograms, and ν is speed in meters per second. The SI unit for kinetic energy, as with all forms of energy, is the joule (J), which is equal to kgm2s-2. From the equation, we can see that the kinetic energy is a function of the square of the speed. If the speed doubles, the kinetic energy will quadruple, assuming the mass is constant. Also note that kinetic energy is related to speed, not velocity. An object has the same kinetic energy regardless of the direction of its velocity vector. Potential energy refers to energy that is associated with a given object’s position in space or other intrinsic qualities of the system.8 Potential energy is often said to have the potential to do work, and can take named forms.

The ball in the first example is not moving and so it is said to possess potential energy. The ball in the second example has been released and now it is moving downwards. The ball now possesses kinetic energy.

Gravitational potential energy depends on an object’s position with respect to some level identified as the datum (the zero potential energy position). The equation that we use to calculate gravitational potential energy is

U = mgh

where U is the potential energy, m is the mass in kilograms, g is the acceleration due to gravity, and h is the height of the object above the datum. The height used in the potential energy equation is relative to whatever the problem states is the ground level. It will often be simply the distance to the ground, but it doesn’t need to be. The zero potential energy position may be a ledge, a desktop, or a platform. Potential energy has a direct relationship with all three of the variables, so changing any one of them by some given factor will result in a change in the potential energy by the same factor. Tripling the height or tripling the mass of the object will increase the gravitational potential energy by a factor of three. Springs and other elastic systems act to store energy. Every spring has a characteristic length at which it is considered relaxed, or in equilibrium.8 When a spring is stretched or compressed from its equilibrium length, the spring has spring potential energy, which can be determined by:

U = ½ kx2

Where U is the potential energy, k is the spring constant (a measure of the stiffness of the spring), and x is the magnitude of displacement from equilibrium. The sum of an object’s potential and kinetic energies is its total mechanical energy. The equation is:

E = U + K

where E is total mechanical energy, U is potential energy, and K is kinetic energy. The first law of thermodynamics accounts for the conservation of mechanical energy, which posits that energy is never created nor destroyed, it is merely transferred from one form to another. This does not mean that the total mechanical energy will necessarily remain constant, though. The total mechanical energy equation accounts for potential and kinetic energies but not for other forms of energy, such as thermal energy that is transferred as a result of friction (heat). If frictional forces are present, some of the mechanical energy will be transformed into thermal energy and will dissipated from the system and not be accounted for by the equation. There is no violation of the first law of thermodynamics, as a full accounting of all the forms of energy (kinetic, potential, thermal, sound, light, etc.) would reveal no net gain or loss of total energy, but merely the transformation of some energy from one form to another.

In the absence of non-conservative forces, such as frictional forces, the sum of the kinetic and potential energies will be constant. Conservative forces are those that are path independent and that do not dissipate energy. Conservative forces also have potential energies associated with them. One method is to consider the change in energy of a system in which the system is brought back to its original setup. In mechanical terms, this means that an object comes back to its starting position. If the net change in energy is zero regardless of the path taken to get back to the initial position, then the forces acting on the object are conservative. This means that a system that is experiencing only conservative forces will be given back an amount of usable energy equal to the amount that had been taken away from it in the course of a closed path. For example, an object that falls through a certain displacement in a vacuum will lose some measurable amount of potential energy but will gain exactly that same amount of potential energy when it is lifted back to its original height, regardless of whether the return pathway is the same as that of the initial descent. Furthermore, at all points during the fall through the vacuum, there will be a perfect conversion of potential energy into kinetic energy, with no energy lost to non-conservative forces such as air resistance. However, non-conservative forces are impossible to avoid in reality. The other method is to consider the change in energy of a system moving from one setup to another. In mechanical terms, this means an object undergoes a particular displacement. If the energy change is equal regardless of the path taken, then the forces acting on the object are again all conservative. When the work done by non-conservative forces is zero, or when there are no non-conservative forces acting on the system, the total mechanical energy of the system (U + K) remains constant. The conservation of mechanical energy can be expressed as

ΔE = ΔU + ΔK = 0

where ΔE, ΔU, and ΔK are the changes in total mechanical energy, potential energy, and kinetic energy, respectively.

Another important quantity related to work is powerP, which is the rate at which work is done or energy is transformed.

P = ΔW/Δt

P =ΔE/Δt

P = force x displacement/time

P= force x velocity

The unit of power is the watt, which is equal to 1 joule per second.

 

References

1) Kane, Thomas R.; Levinson, David A. (1996), Dynamics Online, Sunnyvale, California: Online Dynamics.    vectors

2) Resnick; Halliday; Krane (1992). Physics, Volume 1 (4th ed.). p. 83.

3) Beatty, Millard F. (2006), Principles of Engineering Mechanics, Volume 2: Dynamics—The Analysis of Motion, Mathematical Concepts and Methods in Science and Engineering, 33, Springer

4) Louis Bloomfield, Professor of Physics at the University of Virginia, How Everything Works: Making Physics Out of the Ordinary, John Wiley & Sons (2007)

5) Adkins, C.J. (1968/1983). Equilibrium Thermodynamics, third edition, McGraw-Hill, London

6) Jammer, Max (1957). Concepts of Force. Dover Publications, Inc. p. 167; footnote 14 work

7) C Hellingman (1992). “Newton’s third law revisited”. Phys. Educ. 27 (2): 112–115. Bibcode:1992 PhyEd..27.

8) Resnick & Halliday (1977). Physics (Third ed.). John Wiley & Sons. pp. 78–79. Any single force is only one aspect of a mutual interaction between two bodies.

9) Jain, Mahesh C. (2009). Textbook of Engineering Physics (Part I). p. 9. Chapter 1, p. 9

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