Conservation of Mechanical Energy

The mechanical energy of a system is the sum of its kinetic and potential energies: $E=T+\Pi$. For a system acted on only by conservative forces, this sum remains constant.

Conservation law

If the work of all nonconservative forces is zero, then:

$$T_1+\Pi_1=T_2+\Pi_2=const.$$

Kinetic and potential energy may transform into each other while their sum remains unchanged.

Energy balance

More generally, the change in mechanical energy equals the work of nonconservative forces:

$$E_2-E_1=A_{nc}.$$

For example, negative work of dry friction reduces mechanical energy.

Gravitational system

For a particle moving in a uniform gravitational field without resistance, $mv^2/2+mgh=const$. A decrease in height is accompanied by an increase in kinetic energy.

Elastic system

For a mass attached to an ideal spring with no losses, $mv^2/2+kx^2/2=const$. At extreme positions the speed may be zero while elastic potential energy is maximum.

Choice of potential-energy reference

The conservation law is independent of the chosen zero level of $\Pi$, provided the same reference is used consistently in all states.

Example

A body falls from rest through 5 m without resistance. Taking $\Pi=0$ at the lower level gives $mgh=mv^2/2$, so $v=\sqrt{2gh}\approx9.90$ m/s for $g=9.81$ m/s².

Common mistakes

• applying mechanical-energy conservation while omitting friction work;
• mixing different zero levels of potential energy;
• assuming kinetic and potential energy are separately constant;
• confusing conservation of mechanical energy with conservation of total energy of the physical system.