Work. Line integral. Conservative force fields. Kinetic and potential energy. Conservative and dissipative forces. Power.

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Work. There is in physics a rather abstract concept called “work”. It has a narrow, carefully defined, technical meaning.

In physics, work is done on a body when a force, acting on the body, moves it. The amount of work done is

W = Fs

where F represents the magnitude of the force in the direction of motion and s is the distance moved. See Fig. 1. If the force is not directed in the direction of motion, but at an angle to it, then the work W done by the force is the product of the displacement s and the component of F in the direction of motion. If the force F is pointed at an angle θ to the direction of motion, then W = (F cos θ) s. See. Fig. 2.

The force may not necessarily be constant during the motion, but instead may vary from point to point. The general formula for work is

W = ∫_C F cos θ ds

where the C is the path over which integration is performed. The path C is usually given in the form of parametric equations

x = x(t)

y = y(t)

z = z(t) .

The integral ∫_C F cos θ ds is called a line integral.

The meaning of the word “work” as used in physics is thus much narrower and more technical than its meaning in ordinary, everyday usage. In ordinary usage the word can mean any kind of mental or physical exertion or activity. Simply holding a 10 lb object out at arm’s length for 5 minutes may seem like work in the ordinary sense but it is not work in the physics sense. Carrying a 50 lb bag of potatoes for a mile along a level road may seem like work in the ordinary sense but it is not work in the physics sense since the upward force that you exert on the bag doesn’t produce its horizontal motion.

Units of work

English system. Unit of work is the foot-pound. If one pound of force acts through a distance of one foot, it does one foot-pound of work.

MKS system. Unit of work is the joule. If one newton of force acts through a distance of one meter, it does one joule of work.

CGS system. Unit of work is the erg. If one dyne of force acts through a distance of one centimeter, it does one erg of work.

The term work was introduced in 1826 by the French mathematician Gaspard-Gustave Coriolis as “weight lifted through a height” in connection with the lifting of buckets of water out of flooded mines by early steam engines.

Consider the following. Let us lift 1000 lb of water to the top of a 500 foot tower. In lifting the water to the top of the tower we do 500×1000 = 500,000 ft-lb of work on it and thus impart to it an energy called potential energy equal to the amount of work done in lifting it to that position. Let us now send the water down through a water chute to a water turbine at ground level. When the water comes down the chute its potential energy is converted to an energy of motion called kinetic energy and the water turbine transforms that kinetic energy into electrical energy that can be used to heat buildings, run machines, etc. Let us now take that very same water (that we collected in a pool) and lift it back up to the top of the tower and repeat the process. We thus have a process that we can repeat indefinitely. It is obvious that this system is supplying us usable energy in the form of electrical energy. Where does that energy that we are producing come from? If we assume that perpetual motion machines are impossible and that one cannot create energy from nothing, then the conclusion is inescapable that we have imparted energy to the water by the act of lifting it to the top of the tower. We see in all this the origin and motivation for the concept of work as given by the formula W = Fs.

Conservative force fields. A conservative force field is a force field in which the work done in moving a body from one point to another is independent of the path along which the body is moved. The gravitational field of the earth and the electrostatic field of a point charge are examples of conservative fields.

The work W done in moving a body from one point to another along some path C in xyz space is given by the line integral ∫_C F ⋅ T ds

W =∫_C F ⋅ T ds

where F (a vector) is the force acting on the body and T is the unit tangent to the curve (The dot product F⋅T in the integral represents the tangential component of the force F along curve C). The path C is generally defined parametrically

x = x(t)

y = y(t)

z = z(t) .

Condition for a line integral to be independent of path. Let C be a space curve running from some point P₁ to another point P₂ in some simply connected region Q of space. Let C be defined by the position vector R(s) = x(s) i + y(s) j + z(s) k where s is the distance along the curve measured from point P₁. Let F(x, y, z) = f₁(x, y, z) i + f₂(x, y, z) j + f₃(x, y, z) k represent a force field defined over the region. Let T denote the unit tangent to the curve at point (x, y, z). Then the value of the line integral

∫_C F ⋅ T ds = ∫_C F ⋅ T dR = ∫_C f₁ dx + f₂ dy + f₃ dz

taken along some path (i.e. space curve) C from point P₁:(x, y, z) to point P₂:(x, y, z) within Q will be independent of the particular path chosen if and only if the integrand

f₁ dx + f₂ dy + f₃ dz

is a total (or exact) differential of some function Φ. If the integrand is a total differential then there will exist some function Φ(x, y, z) such that

dΦ = f₁ dx + f₂ dy + f₃ dz .

In this case

where the function Φ(x, y, z) is a scalar point function called the potential function.

The integrand of the line integral will be an exact differential if and only if

This condition is equivalent to curl F = 0.

The idea of the work done in moving a body from one point to another being independent of the path taken can be illustrated by the gravitational field of the earth. The gravitational field of the earth is directed vertically downward toward the center of the earth. If we carry a 50 pound object to the top of a 100 foot hill, the amount of work that we do in carrying it there is 50×100 = 5000 ft-lbs no matter what route we take in going up the hill. We are carrying the object 100 feet against the force of gravity and no matter what route we take, we are still carrying it 100 feet against the force of gravity. The formula for the work done in moving an object from one point to another is W = ∫F cos θ ds where θ is the angle between the direction of the force and the direction of travel. The integrand “F cos θ ds” in this formula can be thought of in two ways:

1) (F cos θ) × ds which corresponds to the dot product F ⋅ T ds definition with the “F cos θ” being the component of F in the direction of travel

2) F×(ds cos θ) where here “ds cos θ” is the component of travel in the direction of the force F

It often gives the best insight to think of it in the second way.

Energy. What is energy? Energy has been defined as the capacity for doing work. An object acquires energy when work is done in raising it to some elevated position. An object also acquires energy when it is set into motion.

There are two kinds of energy in mechanics: Potential energy and kinetic energy. Potential energy is stored energy or energy of position. A bucket of water sitting atop a 500 foot tower has potential energy. The coiled mainspring of a watch has potential energy (work was done in winding the spring). Kinetic energy is the energy of motion. Falling or running water, wind, a speeding bullet all possess kinetic energy.

Units of energy. The units of both potential energy and kinetic energy are the same as for work. They are foot-pounds, joules and ergs.

Potential energy. The potential energy (P.E.) o f a body is the ability of the body to do work because of its position or state.

The potential energy of a mass lifted a vertical distance h is given by the formula

P.E. = mgh

where g is the acceleration due to gravity. Here mg represents the force required to overcome the pull of gravity and lift the object. In the English system this formula becomes

P.E. = wh

where w is the weight.

Kinetic energy. Suppose a force F acts on a body of mass m moving it a distance s, going from a position s₁ to a position s₂ on a frictionless surface. Assume F may be variable and is pointed in the direction of motion. Suppose the speed of the body increases from v₁ at s₁ to v₂ at s₂. See Fig. 3. The work done in moving the mass this distance s is

Now F = ma and if we note that

we can write 1) as

3) W = ½ mv₁² - ½ mv₂²

Note that this expression for the work done does not include either the force F or the displacement s. Only the mass of the body and its original and final speeds appear. The force can vary in magnitude in any manner. The quantity

K = ½ mv²

is called the kinetic energy of a body of mass m moving with speed v.

Relationship between kinetic and potential energies. Consider the double inclined plane of Fig. 4. A marble is placed at the top on the left side. At this point the marble has a potential energy of mgh relative to the bottom of the plane. The marble is released and it rolls down the plane, gaining speed. At the bottom all its potential energy has been transformed into kinetic energy, its speed and kinetic energy are at the maximum. It then goes up the other side. If we assume no friction (no loss of energy due to friction) the ball will roll all the way to the top of the other side where it will come to a momentary stop. At this point it will have lost all its kinetic energy and regained all its potential energy and will again have a potential energy of mgh. It then rolls back down and up the other side and repeats. It will continue rolling back and forth indefinitely if there is no friction. If we assume no friction, at any point in the path the sum of the potential energy and the kinetic energy of the marble is a constant (and that constant = mgh) i.e. at any point in the path

potential energy + kinetic energy = mgh

Thus the total energy of the marble remains constant. This is called the principle of conservation of mechanical energy. It is only valid if there is no loss due to friction and no work is done by outside forces.

The pendulum shown in Fig 5 works in the same way. We have simply replaced the restraint of the inclined plane with a string. When we raise the bob up to a height h, it has a potential energy of mgh. When we release it, it falls with increasing speed until it reaches maximum speed and maximum kinetic energy at the bottom. If there is no frictional loss it will continue all the way up to the same height on the other side where it will come to a momentary stop. At this point all of its kinetic energy will have been transformed back into potential energy and it will again have a potential energy of mgh. It will then swing the other way and continue swinging indefinitely. At any point in its path the sum of the potential energy plus the kinetic energy add to mgh.

Conservative and dissipative forces. Work must be done in lifting a body against the force of gravity. The work that is done is accompanied by an increase in the gravitational potential energy of the body equal to the amount of work done. Thus no energy has been lost — it has simply been transformed into a different kind of energy. Work is also done if we slide a heavy body along a rough horizontal surface. In this case the potential energy does not change and the work done is converted to heat. Why is it that although work was done in both cases, we have an increase in potential energy in the first case and not in the second? Answer: In the first case, the work is recoverable and in the second case it is not. If we slide the body on the horizontal surface back to its original position, instead of recovering the work done on the original displacement, we do more work on the return trip. Thus, if the work can be recovered, there is an increase in potential energy. If it cannot be recovered, there is not. Forces such as those of gravity or the force exerted by a spring, where the work is recoverable, are called conservative forces. Forces like as those of sliding friction are called nonconservative or dissipative forces. Only when work is done against a conservative force is there an increase in potential energy. Only when all the forces are conservative is the mechanical energy of a system conserved.

It might be argued that the heat generated from sliding friction might be used to do useful work. While this is true, only part of it is recoverable. It is not possible to recover all of it (due to the second law of thermodynamics).

Power. Power is the rate at which work is done.

The power at any instant is given by

Since dW = Fds, instantaneous power is also given by

P = Fv

Units of work. The units of work are as follows:

MKS system. One joule per second. Also called a watt.

1 watt = 1 joule/sec

1 kilowatt (kw) = 1000 watts = 1.34 horsepower

CGS system. One erg per second.

English system. One foot-pound per second and one horsepower

1 horsepower (hp) = 550 ft-lb/sec = 746 watts

References

Dull, Metcalfe, Brooks. Modern Physics.

Schaum. College Physics.

Sears, Zemansky. University Physics.

Semat, Katz. Physics.