Work, Power and Energy

Work Done by a Constant Force

Work is the measure of energy transferred to or from an object when a force moves it. Intuitively, lifting a box twice as high takes twice the effort, and pushing with twice the force over the same distance also takes twice the effort. Work therefore grows with both the force and the distance moved. There is one subtlety: only the part of the force that points along the direction of motion does work. Carrying a heavy bag horizontally does not, in this sense, require lifting work against gravity, because gravity acts perpendicular to the motion.

For a constant force $\vec{F}$ producing a displacement $\vec{d}$, work is defined as the dot product of the two vectors, which captures the component of force along the displacement:

$W = \vec{F} \cdot \vec{d} = F\,d\cos\theta$

Here $\theta$ is the angle between the force and the displacement, and $F$ and $d$ are the magnitudes. Work is a scalar quantity with the SI unit joule (J), where $1\text{ J} = 1\text{ N·m}$. The sign of the work depends on the angle. Work is positive when the force has a component along the motion ($\theta$ less than $90°$), zero when the force is perpendicular to the motion ($\theta = 90°$), and negative when the force opposes the motion ($\theta$ greater than $90°$), as friction does. The child article develops these cases with full examples. The key idea to carry forward is that work is a transfer of energy, and the next section extends it to forces that are not constant.

Work by a Variable Force and the Work-Energy Theorem

Many forces change as an object moves. A spring pulls harder the more it is stretched, and a rocket's thrust changes as fuel burns. For such forces, a single multiplication of force by distance is not enough. The method is to imagine the displacement divided into many tiny steps, over each of which the force is effectively constant, and then add up the small amounts of work. Graphically, if the force is plotted against displacement, the work done equals the area under the curve between the start and end positions. A straight-line graph of force against displacement forms a triangle or rectangle, and its area gives the work directly.

This leads to one of the most useful results in mechanics, the work-energy theorem. It states that the net work done on an object equals the change in its kinetic energy, the energy of motion, which is defined in the next section:

$W_{\text{net}} = \Delta K = K_{\text{final}} - K_{\text{initial}}$

The theorem connects two ideas that might otherwise seem separate. Forces do work, and that work changes how fast an object is moving. If the net work is positive, the object speeds up, and if it is negative, the object slows down. The theorem holds for any force, constant or varying, which is why it is so powerful. The child article covers how to evaluate the area for different graph shapes and proves the theorem from Newton's second law.

Kinetic and Potential Energy

Energy comes in forms, and the two most important ones in mechanics are kinetic and potential energy. Kinetic energy is the energy an object possesses because it is moving. For an object of mass $m$ moving at speed $v$, it is

$K = \tfrac{1}{2}mv^2$

measured in joules. The formula shows that kinetic energy grows with the square of speed, so doubling the speed quadruples the energy. This is why braking distances increase so sharply at high speed. Kinetic energy is what the work-energy theorem changes.

Potential energy is energy stored in a system because of the configuration or position of its parts. Work must be done against certain forces to build this store, and the store can later be released to do work. Two cases matter most. Gravitational potential energy near the Earth's surface is $U = mgh$, where $h$ is height above a chosen reference level and $g$ is the gravitational acceleration. Elastic potential energy stored in a stretched or compressed spring of stiffness $k$, displaced by $x$ from its natural length, is $U = \tfrac{1}{2}kx^2$. Lifting an object increases its gravitational potential energy by the work done against gravity, and a stretched spring holds energy equal to the work done in stretching it. Gravitation and the full definition of gravitational potential at large distances are treated in the sibling topic, Gravitation, and the child article limits itself to the near-surface case. The next section shows why this store is so useful: it can be exchanged for kinetic energy and back again without loss, provided no friction intervenes.

Conservation of Mechanical Energy

Mechanical energy is the sum of kinetic and potential energy, $E = K + U$. Energy conservation says that the total energy of an isolated system never changes; it only moves from one form to another. Mechanical energy is conserved in a narrower and very useful sense. It stays constant when the only forces doing work are conservative forces.

A conservative force is one whose work depends only on the starting and ending positions, not on the path taken. Gravity and the spring force are conservative. Lifting a box one metre straight up or along a zigzag path yields the same gravitational work, as long as the start and end heights are the same. A non-conservative force, such as friction or air resistance, does work that depends on the path, and it converts mechanical energy into thermal energy and sound, which are not counted in $E$. When only conservative forces act, the rule is

$K_1 + U_1 = K_2 + U_2$

where the subscripts denote two positions. This single equation solves many problems without tracking forces or accelerations. A pendulum released from rest at a height swings to the same height on the other side, because its potential energy at the top converts fully to kinetic energy at the bottom and back. A frictionless slide delivers a child to the ground with a speed set only by the height of the slide. A spring launches a block with kinetic energy equal to the elastic energy it stored. In real situations friction is always present, and it appears in the equation as energy lost from the mechanical store, so the final mechanical energy is smaller than the initial value. The child article works through these examples and the friction case in full.

Power and Efficiency

Work tells us how much energy is transferred, but it says nothing about how quickly. A small motor and a large engine might lift the same load to the same height, yet the engine does it in a fraction of the time. Power measures this rate. Average power is the work done divided by the time taken:

$P = \frac{W}{t}$

The SI unit of power is the watt (W), where $1\text{ W} = 1\text{ J/s}$. When the force and velocity are changing, the power at a single instant is given by

$P = \vec{F} \cdot \vec{v}$

where $\vec{v}$ is the instantaneous velocity. This is the instantaneous power, and it equals the average power only when the rate is steady. Power is the link between work and the everyday ratings on machines, such as a 60 W light bulb or a kilowatt-rated kettle.

Efficiency describes how much of the energy supplied to a machine emerges as useful output. Because energy is conserved, the useful output can never exceed the total input, and some energy is always lost, typically as heat and sound through friction. Efficiency is defined as

$\eta = \frac{\text{useful output}}{\text{total input}} \times 100\%$

A motor that draws 100 J and delivers 60 J of useful work is 60% efficient. The remaining 40 J is not destroyed; it has been converted into forms that do not perform the intended task. The child article develops average and instantaneous power with worked examples and discusses the efficiency of common machines.