When Mechanical Energy Is Conserved—and When It Is Not

Start with the scope of the energy equation

The phrase “conservation of energy” is broader than “conservation of mechanical energy.” Total energy is conserved, but the simple mechanical-energy equation

$$K_i+U_i=K_f+U_f$$

is valid only when nonconservative forces do no net work on the chosen system.

Conservative forces move energy between kinetic and potential forms

Gravity and an ideal spring are common conservative forces. Their work can be represented by changes in potential energy. When they are the only forces doing work,

$$\Delta(K+U)=0.$$

Kinetic energy can rise while potential energy falls, or vice versa, but their sum stays constant.

Add nonconservative work when needed

If friction, drag, or an external push transfers mechanical energy, use

$$W_{\text{nc}}=\Delta(K+U)$$

or equivalently

$$K_i+U_i+W_{\text{nc}}=K_f+U_f.$$

A negative $W_{\text{nc}}$ means the system’s mechanical energy $K+U$ decreases. Depending on the system, that energy may be converted to another form, such as thermal energy, or transferred out of the system.

Fresh example with friction

A 2.0 kg block starts from rest 5.0 m above a reference level. Its initial gravitational potential energy is

$$U_i=mgh=(2.0)(9.8)(5.0)=98\,\text{J}.$$

Suppose friction does $-30\,\text{J}$ of work by the time the block reaches the bottom, where $U_f=0$. Then

$$0+98-30=K_f,$$

so $K_f=68\,\text{J}$ and

$$v=\sqrt{\frac{2K_f}{m}}=\sqrt{68}\approx8.25\,\text{m/s}.$$

Ignoring friction would incorrectly predict 98 J of kinetic energy.

Choose the system before choosing the formula

The system boundary determines which interactions are treated as internal or external and which forms of energy are included in the system. A force’s classification as conservative or nonconservative does not change simply because the system boundary changes. Conservative interactions can be represented with potential energy, while nonconservative work changes the system’s mechanical energy according to $W_{\text{nc}}=\Delta(K+U)$.