HiSET Science: Work, Energy, and Power

Basic Equation for Work
The equation for work (W) is fairly simple: W = Fd, where F is the force exerted and d is the displacement of the object on which the force is exerted. For the simplest case, when the vectors of force and displacement have the same direction, the work done is equal to the product of the magnitudes of the force and displacement. If this is not the case, then the work may be calculated as W = Fd cos(θ), where θ is the angle between the force and displacement vectors. If force and displacement have the same direction, then work is positive; if they are in opposite directions, however, work is negative; and if they are perpendicular, the work done by the force is zero.
For example, if a man pushes a block horizontally across a surface with a constant force of 10 N for a distance of 20 m, the work done by the man is 200 N-m or 200 J. If instead the block is sliding and the man tries to slow its progress by pushing against it, his work done is -200 J, since he is pushing in the direction opposite the motion. Also, if the man pushes vertically downward on the block while it slides, his work done is zero, since his force vector is perpendicular to the displacement vector of the block.
It is important to note in each of these cases that neither the mass of the block nor the elapsed time is considered when calculating the amount of work done by the man.


Power
Put simply, power is the rate at which work is done. Power, like work, is a scalar quantity. If we know the amount of work, W, that has been performed in a given amount of time, ∆t, then we may find average power, Pav = W/∆t. If we are instead looking for the instantaneous power, there are two possibilities. If the force on an object is constant, and the object is moving at a constant velocity, then the instantaneous power is the same as the average power. If either the force or the velocity is varying, the instantaneous power should be computed by the equation P = Fv, where F and v are the instantaneous force and velocity. This equation may also be used to compute average power if the force and velocity are constant. Power is typically expressed in joules per second, or watts.

Simple Machines
Simple machines include the inclined plane, lever, wheel and axle, and pulley. These simple machines have no internal source of energy. More complex or compound machines can be formed from them. Simple machines provide a force known as a mechanical advantage and make it easier to accomplish a task. The inclined plane enables a force less than the object's weight to be used to push an object to a greater height. A lever enables a multiplication of force. The wheel and axle allows for movement with less resistance. Single or double pulleys allow for easier direction of force. The wedge and screw are forms of the inclined plane. A wedge turns a smaller force working over a greater distance into a larger force. The screw is similar to an incline that is wrapped around a shaft.

Mechanical Advantage
A certain amount of work is required to move an object. The amount cannot be reduced, but by changing the way the work is performed a mechanical advantage can be gained. A certain amount of work is required to raise an object to a given vertical height. By getting to a given height at an angle, the effort required is reduced, but the distance that must be traveled to reach a given height is increased. An example of this is walking up a hill. One may take a direct, shorter, but steeper route, or one may take a more meandering, longer route that requires less effort. Examples of wedges include doorstops, axes, plows, zippers, and can openers.

Levers
A lever consists of a bar or plank and a pivot point or fulcrum. Work is performed by the bar, which swings at the pivot point to redirect the force. There are three types of levers: first, second, and third class. Examples of a first-class lever include balances, see-saws, nail extractors, and scissors (which also use wedges). In a second-class lever the fulcrum is placed at one end of the bar and the work is performed at the other end. The weight or load to be moved is in between. The closer to the fulcrum the weight is, the easier it is to move. Force is increased, but the distance it is moved is decreased. Examples include pry bars, bottle openers, nutcrackers, and wheelbarrows. In a third-class lever the fulcrum is at one end and the positions of the weight and the location where the work is performed are reversed. Examples include fishing rods, hammers, and tweezers.

Wheel and Axle
The center of a wheel and axle can be likened to a fulcrum on a rotating lever. As it turns, the wheel moves a greater distance than the axle, but with less force. Obvious examples of the wheel and axle are the wheels of a car, but this type of simple machine can also be used to exert a greater force. For instance, a person can turn the handles of a winch to exert a greater force at the turning axle to move an object. Other examples include steering wheels, wrenches, faucets, waterwheels, windmills, gears, and belts. Gears work together to change a force. The four basic types of gears are spur, rack and pinion, bevel, and worm gears. The larger gear turns slower than the smaller, but exerts a greater force. Gears at angles can be used to change the direction of forces.

Pulleys
A single pulley consists of a rope or line that is run around a wheel. This allows force to be directed in a downward motion to lift an object. This does not decrease the force required, just changes its direction. The load is moved the same distance as the rope pulling it. When a combination pulley is used, such as a double pulley, the weight is moved half the distance of the rope pulling it. In this way, the work effort is doubled. Pulleys are never 100% efficient because of friction. Examples of pulleys include cranes, chain hoists, block and tackles, and elevators.

Kinetic Energy
The kinetic energy of an object is that quality of its motion that can be related in a qualitative way to the amount of work performed on the object. Kinetic energy can be defined as KE = mv2/2, in which m is the mass of an object and v is the magnitude of its velocity. Kinetic energy cannot be negative, since it depends on the square of velocity. Units for kinetic energy are the same as those for work: joules. Kinetic energy is a scalar quantity.
Changes in kinetic energy occur when a force does work on an object, such that the speed of the object is altered. This change in kinetic energy is equal to the amount of work that is done, and can be expressed as W = KEf – KEi = ∆KE. This equation is commonly referred to as the work-kinetic energy theorem. If there are several different forces acting on the object, then W in this equation is simply the total work done by all the forces, or by the net force. This equation can be very helpful in solving some problems that would otherwise rely solely on Newton's laws of motion.

Potential Energy
Potential energy is the amount of energy that can be ascribed to a body or bodies based on configuration. There are a couple of different kinds of potential energy. Gravitational potential energy is the energy associated with the separation of bodies that are attracted to one another gravitationally. Any time you lift an object, you are increasing its gravitational potential energy. Gravitational potential energy can be found by the equation PE = mgh, where m is the mass of an object, g is the gravitational acceleration, and h is its height above a reference point, most often the ground.
Another kind of potential energy is elastic potential energy; elastic potential energy is associated with the compression or expansion of an elastic, or spring-like, object. Physicists will often refer to potential energy as being stored within a body, the implication being that it could emerge in the future.

Conservative and Non-Conservative Forces
Forces that change the state of a system by changing kinetic energy into potential energy, or vice versa, are called conservative forces.
This name arises because these forces conserve the total amount of kinetic and potential energy. Every other kind of force is considered non-conservative. One example of a conservative force is gravity. Consider the path of a ball thrown straight up into the air. Since the ball has the same amount of kinetic energy when it is thrown as it does when it returns to its original location (known as completing a closed path), gravity can be said to be a conservative force. More generally, a force can be said to be conservative if the work it does on an object through a closed path is zero. Frictional force would not meet this standard, of course, because it is only capable of performing negative work.

One-Dimensional Analysis of Work Done By a Variable Force
If the force on an object varies across the distance the object is moved, then a simple product will not yield the work. If we consider the work performed by a variable force in one dimension, then we are assuming that the directions of the force and the displacement are the same. The magnitude of the force will depend on the position of the particle. In order to calculate the amount of work performed by a variable force over a given distance, we should first divide the total displacement into a number of intervals, each with a width of ∆x. We may then say that the amount of work performed during any one interval is ∆W = Fav∆x, where Fzv is the average force over the interval ∆x. We can then say that the total amount of work performed is the sum of all work performed during the various intervals. By reducing the interval to an infinitesimal length, we obtain the integral:

This integral requires that the force be a known function of x.

Work Performed by a Spring
If we move a block attached to a spring from point xi to point xf, we are doing work on the block, and the spring is also doing work on the block. To determine the work done by the spring on the block, we can substitute F from Hooke's law into our equation for work performed by a variable force, and arrive at this measure: W = k(xi2 – xf2)/2.
This work will be positive if xi2 > xf2, and negative if the opposite is true. If xi = 0 and we decide to call the final position x, then we may change our equation: W = -kx2/2.
It is important to keep in mind that this is the work done by the spring. The work done by the force that moves the block to its final position will be a positive quantity.
Like all simple harmonic oscillators, springs operate by storing and releasing potential energy. The amount of energy being stored or released by a spring is equal to the magnitude of the work done by the spring during that same operation. The total potential energy stored in a spring can be calculated as PE = kx2/2. Neglecting the effects of friction and drag, an object oscillating on a spring will continue to do so indefinitely, since total mechanical energy (kinetic and potential) is conserved. In such a situation, the period of oscillation can be calculated as T = 2pi*sqrt(m/k).