How does the law of conservation of energy apply to mechanical systems?

The law of conservation of energy states that energy cannot be created or destroyed, only transferred.

In mechanical systems, the law of conservation of energy applies by stating that the total amount of energy in a system remains constant, unless energy is transferred in or out of the system. This means that the sum of the kinetic energy and potential energy in a system is always constant.

For example, consider a simple pendulum. At the highest point of its swing, the pendulum has maximum potential energy and zero kinetic energy. As it swings down, the potential energy is converted into kinetic energy, until it reaches the bottom of its swing where it has maximum kinetic energy and zero potential energy. As the pendulum swings back up, the kinetic energy is converted back into potential energy. At all points in the pendulum's swing, the total energy (kinetic plus potential) remains constant.

Another example is a rollercoaster. As the rollercoaster climbs up the first hill, it gains potential energy. As it goes down the hill, this potential energy is converted into kinetic energy, which is then converted back into potential energy as the rollercoaster climbs up the next hill. Again, the total energy in the system remains constant.

In summary, the law of conservation of energy applies to mechanical systems by stating that the total energy in a system remains constant, unless energy is transferred in or out of the system.