Power

FIELDS OF STUDY: Classical Mechanics

ABSTRACT: Power is the rate of work performed or energy consumed. Work, in turn, is the displacement of an object by the application of force. When speaking of electric power, "work" refers to the movement of a charged particle across an electric field. The International System of Units (SI) unit for power is the watt, which is equal to one joule of work done or energy transferred per second. Nonstandard units such as horsepower are sometimes used instead.

PRINCIPAL TERMS

• current: the rate at which an electric charge, usually in the form of electrons, moves through a wire or other conductive material.
• force: a change in the motion of an object caused by its interaction with another object.
• horsepower: an alternative unit of power, in theory based on the amount of power an average horse can produce; commonly used to describe engines.
• mechanical advantage: the amplification of force using a tool, device, or machine, such as a lever or pulley.
• simple machine: a simple mechanical device that redirects or amplifies force.
• voltage: the difference in electric potential between two points, measured in volts; electric current flows naturally from the higher-voltage point to the lower-voltage point.
• watt: the International System of Units (SI) unit of power, equal to one joule of work or energy expended per second.
• work: the displacement of an object as a result of the application of force.

What Is Power?

In physics, power is the rate at which work is performed or energy is consumed over time. The International System of Units (SI) unit for measuring power is the watt (W), which is equal to one joule of energy consumed per second (J/s). Watts are a familiar unit, found on many electric devices to denote their power consumption over time. However, they have a much broader use within physics.

In some instances, particularly when dealing with engines, horsepower may be used instead of watts. Horsepower is a non-SI unit that is based on the ability of an average horse to do work over time. It is poorly standardized, with different types of horsepower equating to different measurements in watts. For example, one metric horsepower is equal to approximately 735.5 watts, one mechanical horsepower is approximately 745.7 watts, and one electric horsepower is exactly 746 watts. The most extreme difference is boiler horsepower, one of which is equal to 9,809.5 watts.

Power, Work, and Energy

Power is simply a measure of work done over time. Work is a way of quantifying the successful application of force or transfer of energy. The relationship between power (P), work (W), and time (t) can be expressed as follows:

To fully understand the factors that go into calculating power, it helps to break down this basic formula into the smallest possible component pieces, beginning with work.

Work is the product of the force (F) acting on an object and the displacement (s) of the object from its original position as a result of that force. (Displacement is the shortest distance between an object’s initial position and its final position, regardless of the path taken between the two.) Assuming that the force in question is the source of all displacement, work can be calculated as follows:

W = Fs

A force, meanwhile, is any interaction that changes the motion of an object—in simple terms, a push or a pull. Force is quantified in terms of the mass (m) of the object experiencing the force and its resulting acceleration (a):

F = ma

Acceleration is a measure of the change in velocity (Δv) of an object over a period of time (t):

These formulas for force and work can be plugged back into the original formula for power, resulting in an equation that defines power in terms of mass, velocity, displacement, and time:

Note that this equation assumes that the force that ultimately generates the power is entirely responsible for the object’s displacement. Real-world calculations involving work, and therefore power, must account for any difference between the direction of the force and the overall direction of movement.

Machine Power

Machines work by taking an applied force and then transforming and transmitting it. The blades of a windmill capture the force of wind blowing across a plain. As the blades turn, a series of connected devices turns the wind force into the grinding force of a millstone.

In addition to redirecting and transforming a force, a machine may also amplify it. The degree to which a machine amplifies applied force is called its mechanical advantage. Devices such as levers, pulleys, and wedges are called simple machines because they are the simplest possible devices that can generate a mechanical advantage. More complicated machines often consist of a series of interconnected simple machines.

It is important to note that the amount of work, and thus power, input into a machine must be the same as its output. Energy can be neither created nor destroyed, and work is simply the transfer of energy. If a machine amplifies a force, the displacement that results from that force must decrease in order for work to remain constant.

No machine is perfect, so real-world machines often have a unique power coefficient to denote what percent of the total available power the machine can actually use. If a machine is said to operate at 80 percent efficiency, it can only use 80 percent of the power available to it. Such a machine would have a power coefficient of 0.8.

Electric Power

Electric power is fundamentally similar to mechanical power, but the terminology is different. Electric power is generated by a current, which is the movement of charged particles (usually electrons) across a conductive material. Currents flow between points with differing electric potentials.

"Electric potential" refers to the electric potential energy that a single charge would have at a given point in an electric field. It is equal to the work that would need to be done to carry the charge to that point when traveling against the field. On its own, the electric potential of a single point is meaningless. It only becomes useful when there is a second point in the electric field for the charge to travel to (or from). The difference between the electric potentials of these two points is called voltage. It is a measure of the energy that will be released if a charge is allowed to travel from the point of higher electric potential to the point of lower electric potential. The SI unit of electric potential is the volt, so the electric potential difference is also measured in volts.

One way of generating electric power is with a hydroelectric dam. Moving water carries kinetic energy, just like any other moving substance, and the wide, flat blades of a water-powered turbine provide a large surface area across which this energy can be captured. In theory, the total power that a turbine could extract from the kinetic energy of moving water—or from any other fluid, such as wind—can be calculated with the formula

where ρ is the density of the water (or other fluid), A is the area of the turbine blade (or other energy-capturing device), and v is the velocity at which the water is flowing. However, like any other machine in the real world, a turbine cannot be 100-percent efficient. To account for the inefficiency of a particular power generator, a power coefficient (C_p) must be added to the above formula:

Again, the power coefficient is a decimal representation of the percent of the total power generated that a machine is able to extract and convert.

Sample Problem

Consider a hydroelectric dam with a water-powered turbine that operates at 60-percent efficiency. The blades of the turbine have a surface area of 5 meters squared (m²). The flow of water through the dam is carefully regulated so that it maintains a constant velocity of 2 meters per second (m/s). Assuming that the water has the average density of fresh water—approximately 1,000 kilograms per cubic meter (kg/m³)—calculate how much energy the dam generates in a single day.

Answer:

The equation for calculating the total power generated by a fluid flow is

Plug in the values given for fluid density (ρ), blade surface area (A), and fluid velocity (v), then solve, paying attention to the units:

The units of kg·m²/s³ may seem troublesome, but one watt is actually equivalent to one kg·m²/s³. Thus, in a perfect world, the total power available to the turbine would be 20,000 watts. To adjust for real-world imperfection, this value must be multiplied by the power coefficient (C_p). If the dam operates at 60-percent efficiency, then its power coefficient is 0.6.

P_usable = C_pP_total

P_usable = (0.6)(20,000 W) = 12,000 W

The dam generates 12,000 watts of power. Because one watt is equal to one joule per second, this means that 12,000 joules of energy are being produced each second. To calculate how much energy is produced in one day, simply multiply this amount by the number of seconds in a day:

(24 h)(60 min/h)(60 s/min) = 86,400 s

E_daily = (12,000 J/s)(86,400 s)

E_daily = 1,036,800,000 J

The dam generates 1,036,800,000 joules, or 1.0368 gigajoules (GJ), of energy per day.

Power All Around

Although the average individual is unlikely to need to calculate the energy output of a hydroelectric dam, understanding the principles of how power is calculated is tremendously useful. Electric power in particular is a vital part of everyday life, with electric bills accounting for a major household expense. Consumers should pay close attention to the wattage and power ratings of various appliances. The principles of power, work, force, and energy also underpin all machines, from the simple wheel-and-axle arrangement of a doorknob to the more complicated components found in combustion engines or washing machines.

Bibliography

"Estimating Appliance and Home Electronic Energy Use." Energy.gov. US Dept. of Energy, 10 May 2015. Web. 23 July 2015.

Henderson, Tom. "Vectors: Motion and Forces in Two Dimensions." The Physics Classroom. Physics Classroom, n.d. Web. 23 July 2015.

International Committee for Weights and Measures. The International System of Units (SI). Updated 8th ed. Sevres: BIPM, 2014. Bureau International des Poids et Mesures. Web. 23 July 2015.

"Kinematics." Encyclopaedia Britannica. Encyclopaedia Britannica, 6 June 2013. Web. 23 July 2015.

Siegel, Ethan. "What Does Torque in a Car Do?" Starts with a Bang! ScienceBlogs, 21 Apr. 2009. Web. 23 July 2015.

Simanek, Donald E. "Kinematics." A Brief Course in Classical Mechanics. Lock Haven U, Feb. 2005. Web. 23 July 2015.