Work and Energy: The Work-Energy Theorem Explained

In everyday language, "work" can mean almost any kind of effort. In physics, the definition is much more precise. Work is done on an object only when a force causes that object to move some distance in the direction of the force. If you push against a wall with all your strength and it does not budge, you have done zero work on the wall, even though you are exhausted.

The physics equation for work is:

\( W = F d \cos\theta \)

where \(W\) is work (measured in joules, J), \(F\) is the applied force (in newtons), \(d\) is the displacement of the object (in meters), and \(\theta\) is the angle between the force vector and the direction of displacement.

Two special cases are worth remembering. When the force acts in the same direction as the motion (\(\theta = 0\)), \(\cos\theta = 1\) and the formula simplifies to \(W = Fd\). When the force acts perpendicular to the motion (\(\theta = 90^\circ\)), \(\cos\theta = 0\), so the force does no work at all, no matter how strong it is.

Two forces show up constantly in these problems, so it helps to know their work formulas directly.

The work done by the gravitational force as an object falls or rises a vertical height \(h\) is \(W_g = mgh\), where \(m\) is mass and \(g\) is the acceleration due to gravity. Gravity does positive work when an object moves downward and negative work when an object moves upward, since the force and displacement point in opposite directions in that case.

The work done by friction is \(W_f = -F_f d = -\mu F_N d\), where \(F_f\) is the friction force, \(\mu\) is the coefficient of friction, and \(F_N\) is the normal force. Friction almost always opposes motion, so the work it does on a moving object is negative: it removes kinetic energy from the system rather than adding to it.

Energy is the capacity to do work, and one of the most useful forms is kinetic energy, the energy an object has because it is moving:

\( KE = \frac{1}{2}mv^2 \)

The graph below shows how kinetic energy grows with speed for a fixed mass. Because \(v\) is squared, doubling an object's speed quadruples its kinetic energy, not just doubles it.

The work-energy theorem connects the two ideas: the net work done on an object equals its change in kinetic energy.

\( W_{net} = \Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2 \)

This single equation, sometimes called the work-kinetic energy theorem or the work-energy principle, is one of the most tested ideas in an introductory mechanics course because it lets you solve motion problems using energy instead of tracking acceleration and time separately.

A 2 kg box starts at rest and is pushed 4 m across a floor by a horizontal force of 10 N. Friction exerts a constant 3 N force opposing the motion. Find the box's final speed.

First find the net force: \(F_{net} = 10\) N\( - 3\) N\( = 7\) N\(\).

Next find the net work: \(W_{net} = F_{net} d = 7\) N\( \times 4\) m\( = 28\) J\(\).

By the work-energy theorem, \(W_{net} = \Delta KE\). Since the box starts at rest, \(\Delta KE = \frac{1}{2}mv_f^2\), so:

\( 28 = \frac{1}{2}(2)v_f^2 \)

Solving gives \(v_f^2 = 28\), so \(v_f \approx 5.3\) m/s.

Work and the work-energy theorem describe how forces transfer energy into or out of an object's motion, but they are only part of the story. When forces like gravity and springs are involved, it is often easier to track energy using conservation of energy, which follows kinetic and potential energy together as they trade back and forth. And once you start asking how quickly work gets done, that leads to power and efficiency, which measures work per unit time rather than work alone.

Mastering the work formula and the work-energy theorem now will make both of those later topics much easier to pick up, since they all share the same underlying idea: forces doing work to move energy from one form, or one object, to another.

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