Mechanics: Chapter 13 – Conservation of Mechanical Energy

When a force is conservative, the work done by the force on an object equals the change in the potential energy of the object. If the work done is positive, there is a decrease in the potential energy of the object. Similarly, if the work done is negative, there is an increase in the potential energy of the object. Hence,

  • W = - \Delta U

Energy is always conserved. Essentially the sum of the kinetic energy K and the potential energy U of a system is constant.

  • K + U = c where c is a constant

This also implies that if no external energy is added or removed from the system, the initial energy E_{i} of a system is equal to the final energy E_{f} of a system, and hence

  • K_{i} + U_{i} = K_{f} + U_{f}

This is easy to prove in the case of a conservative force like gravity. Let’s say an object is dropped from rest in free fall, now the work done, W by gravity on the object is, as I’ve already established,

  • W = - \Delta U

and since the force of gravity accelerates the object from rest to a speed v, the work done by the force of gravity on the object increases the Kinetic energy of the object. In other words,

  • W = \Delta K

Now

  • \Delta K = - \Delta U \Rightarrow \Delta K + \Delta U = 0

If the initial Kinetic energy and potential energy of the system were K_{i} and U_{i} and

  • K_{i} + U_{i} = c

Then the final kinetic and potential energies, K_{f} and U_{f} of the system can be written as the sum of the initial energies and the change in energies. Hence,

  • K_{f} + U_{f} = (K_{i} + \Delta K) + (U_{i} + \Delta U) = (K_{i} + U_{i}) + (\Delta K + \Delta U)

We know that \Delta K + \Delta U = 0,  hence

  • K_{i} + U_{i} = K_{f} + U_{f}

Quod erat demonstrandum.

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