Work, energy and power

For a constant force, work depends on the component parallel to the displacement: $W=Fs\cos\theta$, where $\theta$ is the angle between the force and displacement vectors. Work is a scalar and its SI unit is the joule: $1\,\mathrm{J}=1\,\mathrm{N\,m}$.

Work is positive when the parallel force component points with the displacement, negative when it points against the displacement, and zero when it is perpendicular. A normal force on an object sliding along a fixed surface therefore does no work because the force and displacement are at $90^\circ$.

If force changes with position, split the motion into narrow displacement intervals. The signed area under a force–displacement graph gives the work: area above the displacement axis is positive and area below it is negative. Rectangles, triangles, and trapezia often provide the required area without calculus.

Translational kinetic energy is $E_k=\tfrac12mv^2=\frac{p^2}{2m}$, where $p=mv$ is the magnitude of momentum. It is scalar, cannot be negative, and depends on speed rather than velocity direction. Doubling speed multiplies kinetic energy by four; doubling mass at fixed speed only doubles it.

The work–energy theorem states that the net work done by all external forces equals the change in kinetic energy: $W_{\mathrm{net}}=\Delta E_k$. Positive net work increases speed, while negative net work decreases speed.

Net work is not generally the work done by just one convenient force. If a driving force, friction, gravity, and a normal force act, add the work done by every force or first find the resultant force component along the displacement.

Near Earth's surface, a change in height $\Delta h$ changes gravitational potential energy by $\Delta E_g=mg\Delta h$. The result depends only on the initial and final heights, not on the path between them, because gravity is a conservative force.

The zero of gravitational potential energy is a chosen reference level. Changing the reference changes the numerical value of $E_g$, but it does not change $\Delta E_g$ or any prediction. State the reference when an absolute value is shown.

Gravity does positive work when an object moves downward and negative work when it moves upward: $W_g=-\Delta E_g$. When gravity is the only force doing work, a decrease in $E_g$ appears as an equal increase in kinetic energy.

Within its linear elastic range, a spring obeys Hooke's law: the restoring force is opposite to displacement and has magnitude $kx$. The extension or compression $x$ is measured from the spring's natural, unstrained length.

Because spring force grows from zero to $kx$, the work required to deform it is the triangular area under its force–extension graph. The elastic potential energy stored is therefore $E_s=\tfrac12kx^2$, not $kx$ or $kx^2$.

When the spring returns toward its natural length, it can transfer elastic potential energy into kinetic energy or another store. The quadratic energy model requires a linear force–deformation relationship. An object can still be elastic beyond its limit of proportionality; beyond its elastic limit, it may not return fully to its original shape.

Mechanical energy is the combined kinetic and potential energy of the chosen system: $E_{\mathrm{mech}}=E_k+E_g+E_s$. If only conservative forces transfer energy within an isolated mechanical system, this total remains constant.

An energy equation compares states rather than describing the route between them. List every relevant store at the initial and final states, include any transfer across the system boundary, and cancel terms that are zero or unchanged. Mass often cancels in gravity-only motion, explaining why the predicted speed is independent of mass.

Schematic snapshots, not numerical scale drawings. On the ideal track, kinetic and gravitational potential energy exchange while their sum remains constant. With friction, some mechanical energy becomes thermal energy. Each full bar represents the same total for the cart–track–Earth system, assuming negligible energy transfer to its surroundings. The first and last snapshots still have kinetic energy: they are not release-from-rest or turning points. The right-hand cart is shown at one lower uphill position, not at a calculated maximum height. The gravitational potential-energy reference is arbitrary, so its valley value need not be zero.

Friction and drag can do negative work on a moving object. When they reduce its mechanical energy, that energy has not disappeared: it is transferred mainly into internal or thermal energy of the object and surroundings, and sometimes sound. Friction from a moving belt or drag from moving air can instead do positive work on the chosen object; identify the system and the direction of each transfer.

For kinetic friction of approximately constant magnitude, the thermal transfer is $\Delta E_{\mathrm{th}}=f_k s$. More generally, the work done by non-conservative forces equals the change in mechanical energy of the selected system.

Braking is a direct work–energy application. With the same mass and braking force, $Fd=\tfrac12mv^2$, so braking distance is proportional to speed squared. Doubling speed therefore requires about four times the braking distance, before adding the driver's reaction distance.

Average power is energy transferred per time interval: $P_{\mathrm{avg}}=\frac{\Delta E}{\Delta t}$. One watt is one joule per second. Two machines can perform the same work but the machine that completes it sooner has greater average power.

For a force acting on an object moving with instantaneous velocity, mechanical power is $P=\vec F\cdot\vec v=Fv\cos\theta$. Only the force component parallel to velocity transfers energy at that instant.

At constant speed on a level road, the driving force balances the total resistance, so engine output power at the wheels is $P=F_{\mathrm{res}}v$. At changing speed, distinguish the driving force from the resultant force because some engine power changes kinetic energy while some is dissipated.

Efficiency is the fraction of input energy or power delivered in the useful output pathway. It is dimensionless and can be written as a decimal or multiplied by $100\%$. The numerator and denominator must describe the same time interval and use matching energy or power quantities.

What counts as useful depends on the stated purpose. Light from a lamp may be useful while thermal transfer is dissipated; for a heater, the thermal transfer is the intended useful output. Always identify the function before labelling pathways.

In a Sankey diagram, arrow width represents the amount of energy or power in each pathway. For a device with no net energy storage change, the input arrow splits into useful and dissipated outputs whose combined widths equal the input width. If energy is being stored or released, include that change in the energy balance rather than assuming the output alone equals the input.

Real systems often combine several stores and transfers: a skier loses gravitational potential energy while gaining kinetic and thermal energy; a hydroelectric station turns a gravitational transfer into electrical output; a motor may lift a load while warming its surroundings. For a fuel, volumetric energy density is $u=\frac{E_{\mathrm{fuel}}}{V}$, in $\mathrm{J\,m^{-3}}$; specific energy is $e=\frac{E_{\mathrm{fuel}}}{m}$, in $\mathrm{J\,kg^{-1}}$. Hence $E_{\mathrm{fuel}}=uV=em$ and $u=\rho e$, where $V$ is fuel volume, $m$ its mass, and $\rho=\frac{m}{V}$ its mass density. Useful output is $\eta E_{\mathrm{fuel}}$. Compare fuels on the same mass or volume basis: greater energy density reduces the volume needed for a given energy, but does not by itself imply greater efficiency.

For a multistage problem, compare the first and last states whenever intermediate details are unnecessary. Then add any known dissipated transfer. This energy route is often shorter than calculating acceleration and time separately, but it does not directly provide direction or trajectory.

For a simple cascade of devices, suppose each stage receives exactly the useful output of the previous stage, with no extra input or bypass pathway. The overall efficiency is then the product of the stage efficiencies as decimals, which explains why many moderately efficient stages can produce a much lower total efficiency.

A spring investigation measures the applied stretching-force magnitude for several extensions, plots that force on the vertical axis against extension on the horizontal axis, and uses the best-fit gradient for the spring constant. The area under this graph gives the work stored elastically, provided the loading remains within the linear elastic region. A graph of the signed restoring force instead has gradient −k.

A human-power investigation measures mass, vertical height, and climbing time, then estimates useful power from the gravitational energy gained per time. Repeated timing reduces random variation; a clearly defined start and finish and a calibrated height measurement reduce systematic error.

For exam calculations, sketch the initial and final states, choose the system and zero level, list energy stores and transfers, write a symbolic conservation equation, substitute SI values, and finish with units and a reasonableness check. Do not start by hunting for a formula without an energy model.