3 Work and Energy

Work and displacement

Work describes energy transferred by a force while an object undergoes displacement. For a constant force $\vec F$ and displacement $\Delta\vec r$,

where $\theta$ is the angle between the force and displacement. Only the component of force parallel to the displacement contributes. Work is positive if that component points along the displacement, negative if it points opposite the displacement, and zero if the force is perpendicular.

For example, a $10\,\text{N}$ force pulling at $60^\circ$ over $3\,\text{m}$ does

When force varies along the path, sum its contributions over small displacements by integration:

For one-dimensional motion, $W=\int_{x_A}^{x_B}F_x(x)\,dx$. This integral is the signed area under a force-versus-position graph. For instance, if $F_x=2x\,\text{N}$ from $x=0$ to $x=3\,\text{m}$, then $W=\int_0^3 2x\,dx=9\,\text{J}$.

Takeaway: Work depends on displacement and on the force component along that displacement.

Work changes kinetic energy

An object's Kinetic energy is the energy associated with its motion. For mass $m$ and speed $v$,

The Work–energy theorem connects forces to changes in motion: the net work done by all forces on an object equals its change in kinetic energy.

Net work includes contributions from every force acting on the object. Positive net work increases kinetic energy; negative net work decreases it. This relationship can be useful for finding a change in speed even when acceleration varies.

Example: If the net work on an object is positive, its final kinetic energy must be greater than its initial kinetic energy. This does not by itself identify which force contributed most; the net work is the combined effect of all forces.

Takeaway: To relate the work of all forces to speed, use the change in kinetic energy.

Potential energy and conservative forces

Potential energy describes energy associated with interactions within a chosen system. It is defined for conservative forces, whose work between two positions depends only on those positions, not on the path between them. The change in potential energy is the negative of the work done by the conservative force:

Near Earth's surface, gravitational potential energy can be written as

where $h$ is measured from a chosen zero-height reference. For an ideal spring displaced by $x$ from its relaxed length, spring potential energy is

where $k$ is the spring constant. The reference level for potential energy can be chosen for convenience; changes in potential energy are what matter for predictions.

Takeaway: Conservative forces allow changes in stored potential energy to be related directly to their work.

Mechanical energy and conservation

Conservation of energy connects kinetic and potential energy. Mechanical energy is their sum:

If nonconservative forces, such as friction, do no work, mechanical energy remains constant:

For example, an object released from rest $5\,\text{m}$ above the ground converts gravitational potential energy into kinetic energy when air resistance is neglected. Equating the initial gravitational potential energy with the final kinetic energy gives

If nonconservative forces do work, include that work in the energy accounting:

Friction can reduce mechanical energy by transforming it into other forms, such as thermal energy. Total energy is conserved when those forms are included in the system.

Takeaway: Mechanical energy is conserved only when nonconservative forces do no work; otherwise, account for their work and any other relevant energy forms.

Power: the rate of energy transfer

Power measures how quickly work is done or energy is transferred. Average power over a time interval is

and instantaneous power is $P=dW/dt$. For a force acting on an object moving with velocity $\vec v$,

The SI unit of power is the watt, equal to $1\,\text{J/s}$. A machine that performs the same work in less time delivers greater average power.

Takeaway: Work describes an amount of energy transferred; power describes the rate of that transfer.