Introduction to Work, Energy and Power

Work, energy and power are among the most widely used ideas in the whole of physics, because they let us describe how one form of motion or configuration changes into another without having to track every force in full vector detail at every instant. This unit builds these ideas up carefully, in the order WBCHSE's own syllabus lays them out.

Work measures the effect of a force acting through a displacement -- it converts a force (a vector) and a displacement (also a vector) into a single scalar number that tells us how much a force actually contributed to changing a body's motion. Work is first defined for the simple case of a constant force, and then extended, using integration, to the more general and more common case of a force whose magnitude (and sometimes direction) changes as the body moves -- friction that varies with normal reaction, a spring's restoring force that grows with extension, or a rocket's thrust that changes as fuel burns away.

Kinetic energy, $K = \tfrac12 mv^2$, is the energy a body has purely because it is moving, and the work-energy theorem connects it directly to work: the net work done on a body by every force acting on it, added together, always equals exactly the change produced in that body's kinetic energy. This single theorem is often a far faster route to a body's speed than working out its full equation of motion and integrating it directly, particularly when the force involved is not constant.

Potential energy is energy stored by virtue of position or configuration rather than motion -- a raised object, a stretched or compressed spring -- and it can be defined only for a conservative force, one whose work does not depend on the path taken, only on the start and end points. Non-conservative forces, chiefly friction and other forms of resistance, cannot store energy this way; instead they dissipate mechanical energy irreversibly, usually as heat.

Putting kinetic and potential energy together as the total mechanical energy, $E = K + U$, leads to the law of conservation of mechanical energy: whenever only conservative forces do work on a body, $E$ stays exactly constant throughout its motion, even as $K$ and $U$ continually trade off against each other. This unit applies that law to motion in a vertical circle, where gravity continually does work on a body moving around a circular path, changing its speed at every point, and finds the minimum speeds needed, at the top and bottom of the circle, for a string to stay taut or for water in a whirled bucket not to spill.

Finally, the unit turns to collisions, where two (or more) bodies interact briefly and strongly. Momentum is always conserved in a collision, but kinetic energy is conserved only in the idealised, limiting case of a perfectly elastic collision; every real collision loses some kinetic energy to heat, sound and permanent deformation, and is therefore, to some degree, inelastic. Both head-on (one-dimensional) and oblique (two-dimensional) collisions are treated, along with the coefficient of restitution, the single number that measures exactly how elastic or inelastic a given collision actually is. Power, the rate at which work is done or energy transferred, is introduced alongside work as the natural next question once work itself is understood: not just how much work is done, but how quickly.