The work-energy theorem connects two fundamental concepts in physics: work and kinetic energy. Work is defined as force times displacement times cosine of the angle between them. Kinetic energy is one-half mass times velocity squared. When a force acts on an object and causes displacement, work is done, which changes the object's kinetic energy.
The work-energy theorem states that the net work done on an object equals the change in its kinetic energy. We can derive this from Newton's second law. Starting with F equals ma, work equals force times distance. Using kinematics, we substitute for acceleration and arrive at the fundamental relationship: work equals the change in kinetic energy.
Let's apply the work-energy theorem to a practical example. A car with mass 1200 kilograms is accelerated by a force of 3000 newtons over a distance of 50 meters, starting from rest. First, we calculate the work done: force times distance equals 150,000 joules. Using the work-energy theorem, this work equals the change in kinetic energy. Since initial velocity is zero, we solve for final velocity and get 15.8 meters per second.
When multiple forces act on an object, we must consider the work done by each force separately. For a block sliding down a rough inclined plane, gravity does positive work, friction does negative work, and the normal force does zero work since it's perpendicular to motion. The net work equals the sum of work done by all forces, which then equals the change in kinetic energy according to the work-energy theorem.
Advanced applications of the work-energy theorem include variable forces like springs, where force equals k times displacement, and work involves integration. In roller coasters, we analyze energy transformations between kinetic and potential energy. The theorem extends to conservation of energy principles, where work done by non-conservative forces equals the change in total mechanical energy. These concepts apply to projectile motion, oscillations, and many real-world engineering problems.