1. Define the system. 2. Identify energy types. 3. Account for work done on/by the system. 4. Apply conservation of energy: $ME_i + W = ME_f$

1. Identify initial and final states. 2. Determine energy forms at each state. 3. Account for any work done. 4. Set up and solve the conservation of energy equation.

Friction converts mechanical energy into thermal energy, reducing the total mechanical energy of the system.

The total energy of the system increases.

The total energy of the system decreases.

Increasing the height increases the gravitational potential energy: $$PE_g = mgh$$

Increasing the spring constant increases the potential energy stored in the spring: $$PE_s = \frac{1}{2}kx^2$$

The total mechanical energy of the system stays constant.

In a closed system, the total energy remains constant. Energy can transform from one form to another, but it cannot be created or destroyed.

Energy of motion. Mathematically represented as $K = \frac{1}{2}mv^2$.

Stored energy due to position or configuration. Examples include gravitational potential energy ($U_g = mgh$) and spring potential energy ($U_s = \frac{1}{2}kx^2$).

Forces for which the work done depends on the path taken. Examples include friction and air resistance.

Forces for which the work done is independent of the path taken. It only depends on the initial and final positions. Examples include gravity and spring force.

The sum of kinetic energy (K) and potential energy (U): $ME = K + U$