Work and Energy

Work is a transfer, and the angle is doing the work

Work is energy moved into or out of a body by a force that acts while the body is displaced. For a force that stays constant over a straight displacement, the amount transferred is W = F d cos(theta), where F is the magnitude of the force, d is the distance moved, and theta is the angle between the force and the displacement. The cos(theta) is not decoration. It selects the component of the force that lies along the motion, and everything counterintuitive about work follows from it.

Three consequences are worth stating flatly, because they are where beginners lose marks. A force perpendicular to the motion does zero work: cos(90 degrees) is zero, so the transfer is zero no matter how large the force. The normal force on a body sliding along a level surface therefore does no work at all, since it points at right angles to the displacement it accompanies. And holding a heavy object still does no work in the mechanical sense, however tiring it is, because the displacement is zero and there is nothing for the force to act through. Your muscles are consuming chemical energy the whole time, and that is a real physiological cost, but it is not work done on the object. Nothing is being transferred to the box you are gripping.

Sign matters too. A force generally along the displacement transfers energy in, and the work is positive. A force generally opposing the displacement transfers energy out, and the work is negative; friction on a sliding body is the standard case. When the force varies along the path, or the path is curved, the constant-force formula stops applying and you integrate the infinitesimal transfers instead: W = integral of F . dr along the path from A to B. That integral is the general definition; W = F d cos(theta) is the special case where the integrand is constant.

Kinetic energy and the work-energy theorem

Kinetic energy is the energy a body has by virtue of moving: K = (1/2) m v squared. It is a scalar, it is never negative because both the mass and the square of the speed are non-negative, and it depends on the reference frame you measure the speed in, since velocity itself is relative.

The work-energy theorem connects the two ideas: the net work done on a particle equals the change in its kinetic energy, W_net = K_final - K_initial. It is not an independent law. It falls out of Newton's second law once you dot both sides with the displacement and integrate, which is why it never tells you anything Newton's second law could not, and why it is often far quicker to use.

The word doing the work in that sentence is net. You must include every force acting on the body, not the ones you find interesting. Leave out friction, or the component of weight along a slope, and the theorem returns a confidently wrong answer. Notice also what the theorem does not contain: time. It relates speeds to distances, so when a problem asks how fast and not when, the energy route usually skips a whole integration. When the problem does ask when, you need the kinematics, which belongs to the dynamics topic.

Potential energy, and the datum you are free to choose

For some forces you can stop tracking work along the path and instead assign a stored energy to the configuration of the system. The potential energy difference between two points is defined as the negative of the work that force does between them: U_B - UA = -W(A to B). Near Earth's surface the gravitational case gives U = m g y plus a constant; for a spring obeying Hooke's law it gives U = (1/2) k x squared plus a constant, where x is the stretch or compression measured from the unstretched length, not the length of the spring.

Those additive constants are the point, not a technicality. Only differences in potential energy are physically meaningful. The zero level, the datum, is something you choose for convenience: floor level, the tabletop, the unstretched spring, the lowest point of the swing. Choose it, write it down, and use the same one on both sides of your equation. A student who asks what the potential energy of a book on a desk is has asked an incomplete question; the answer is a number only once a datum is named. Two students who pick different datums will get different values of U at every point and identical values of every difference, which is the only thing either of them can measure.

One more precision. Potential energy is a property of the interaction between objects in a chosen system, not a possession of a single object. The book does not hold gravitational potential energy on its own; the book-and-Earth system does. Saying it the loose way is harmless until you start drawing system boundaries, at which point it stops being harmless.

Conservative versus non-conservative forces: path independence is the test

You can only build a potential energy function for a force whose work does not depend on the route taken. That is the definition of a conservative force: the work it does between two points is the same along every path connecting them, which is equivalent to saying the work it does around any closed path is zero. Gravity qualifies. An ideal spring qualifies. For these, the work depends only on the endpoints, so a single stored-energy function indexed by position captures everything.

Friction does not qualify, and the reason is easy to see. Kinetic friction always opposes the motion, so it always does negative work, and the amount is proportional to the distance actually traveled. Take a longer route between the same two points and you pay more. Loop back to where you started and the work around that closed path is not zero but negative, which is exactly the closed-path test failing. Air drag behaves the same way. Forces like these are called non-conservative or dissipative, and they have no potential energy function; there is always a conservative force behind every potential energy, and never a dissipative one.

Path independence is the defining test, not a property you check afterwards. If someone hands you an unfamiliar force and asks whether it is conservative, the question to ask is not whether it looks like gravity but whether its work around a closed loop vanishes.

Conservation of mechanical energy, and where friction's share went

Mechanical energy is the sum E = K + U. When no non-conservative force does work on a system, that sum does not change: K + U stays constant, and a problem reduces to setting the total at one instant equal to the total at another. This is why a block released on a frictionless track reaches the bottom at a speed fixed by the drop height alone, whatever shape the track takes in between.

When non-conservative forces do act, the statement generalizes rather than failing: the change in mechanical energy equals the work done by the non-conservative forces, W_nc = delta(K + U). For sliding friction that work is negative and equal in magnitude to the friction force times the path length, so mechanical energy falls by exactly that amount.

Say dissipated, not lost. The energy that leaves the mechanical account has gone into the thermal energy of the rubbing surfaces and the surrounding air; it is somewhere, it raised a temperature, and in principle you can measure it. Calling it lost invites students to believe energy conservation has exceptions. Once you widen the system boundary to include the thermal store, the books balance again. Doing that widening properly, with heat and internal energy as named terms, is the business of the first law of thermodynamics topic, and this lesson deliberately stops at the edge of it. Impulse and the momentum route to collision problems belong to the momentum topic for the same reason: energy and momentum are different accounts, and a collision that conserves one may not conserve the other.

Power, efficiency, and why torque is not measured in joules

Power is the rate at which energy is transferred: average power is the work divided by the interval it took, and instantaneous power is P = dW/dt. Differentiating the definition of work gives the form engineers reach for most often, P = F . v, force dotted with velocity. A motor pulling at 1.60 kN on a load moving at 5.6 m/s is delivering about 9.0 kW at that instant, and the same motor delivered less earlier in the pull when the load was slower.

Efficiency is useful output divided by total input, expressed as a fraction or a percentage. For any real machine it is below one, because some of the input is dissipated; an efficiency above 100 percent would violate conservation of energy, so a claim of one is a claim that something has been miscounted or left out. Be honest about the qualifier, though. A quoted efficiency depends entirely on the boundary drawn around the system and on what the quoter decided counts as useful. The EPA's guidance on combined heat and power makes the point in the open: the same plant yields different efficiency figures under total system efficiency, which counts electric and thermal output alike, and under effective electric efficiency, which credits the thermal output against the fuel a separate boiler would have burned. Neither number is dishonest and neither is comparable to the other. When you read an efficiency, read the boundary first.

The units deserve a sentence of their own. The joule is the SI coherent derived unit of energy and work, equal to one newton-meter and to one kilogram meter squared per second squared. The watt is one joule per second. Now the convention students routinely misread: torque, or moment of force, has exactly the same dimensions as energy, and yet by SI convention it is expressed in newton-meters and the joule is never used for it. The units are dimensionally identical; the quantities are not the same kind of thing, and keeping the names distinct is how the SI signals that. Write N.m for a torque and J for an energy, every time.

A closing note on scope. This lesson is educational material, not engineering design guidance. Nothing here, including the assumed friction coefficient in the worked example, should be used to size a hoist, a cable, a motor, a brake, or any other real component. Real design is performed by a licensed engineer working to the governing code or standard, with verified material properties and margins chosen for the consequences of failure.