Fundamentals (Work and Energy)

Scalar definition of work as a line integral of force along a displacement path.

General idea of the topic

The work of a force is the energy it transfers to a body when the body is displaced. Only the component of the force along the displacement counts: a force perpendicular to the motion does no work, and work can be positive, negative or zero.

The topic defines the work of a constant force, extends it to forces that vary with position by means of integrals or areas under the force-position graph, and defines power as the rate at which work is done.

What you will truly understand

By the end you will be able to compute the work of a constant force with the correct angle, that of a variable force such as the elastic force and the average and instantaneous power of a machine.

You will also be able to interpret the sign of the work: positive if it adds energy and negative if it removes it.

Recommended learning route

1. Work: W = F·d·cos θ, sign and units in joules.
2. Work by a Variable Force: integral of F dx and area under the curve for springs and position-dependent forces.
3. Work-Energy Theorem: the net work on a body equals the change in its kinetic energy.
4. 
   : P = W/t and P = F·v, with the watt and efficiency.

Physical criterion you must acquire

The essential criterion is to identify the angle between force and displacement, not the angle of the plane or of the axes. The normal force and the centripetal force do no work because they are perpendicular to the motion. The maximum power of a machine limits how fast it can transfer energy, not the total amount.

Connection with real problems

• lifting loads.
• motors and electricity consumption.
• cycling and sport.
• springs and shock absorbers.

Final goal of the topic

The final goal is to compute the work and power of any force, constant or variable, with confidence and interpret its sign physically.