The force exerted by an ideal spring is proportional to the displacement from its equilibrium position: (F_s = -k Delta x)

A measure of a spring's stiffness; a larger k means the spring is harder to stretch.

The energy stored in a spring due to its deformation: (U_s = frac{1}{2}k(Delta x)^2)

A force where the work done is independent of the path taken; it only depends on the initial and final positions.

The change in potential energy due to a change in height in a uniform gravitational field: (Delta U = mgDelta h)

The gravitational potential energy between two masses at a distance r: (U = -frac{Gm_1m_2}{r})

1. Identify the conservative force. 2. Determine the initial and final positions. 3. Use the formula (W = -Delta U) or (W = int_{a}^{b} F cdot dr).

1. Determine the spring constant, k. 2. Measure the displacement from equilibrium, Δx. 3. Use the formula (U_s = frac{1}{2}k(Delta x)^2).

1. Determine the potential energy function, U(x). 2. Find the slope of the graph at the point of interest. 3. Take the negative of the slope to find the force: (F = -frac{dU}{dr}).

1. Start with the definition of work: (W = int_{r_i}^{r_f} F(r) dr). 2. Substitute Newton's Law of Universal Gravitation: (F(r) = frac{Gm_1m_2}{r^2}). 3. Integrate to find the change in potential energy. 4. Set initial potential energy to 0 at infinity.

Elastic potential energy is stored in the spring.

There is a negative change in potential energy: (W = -Delta U).

The system is in equilibrium (no net force).

Its gravitational potential energy increases: (Delta U = mgDelta h).

The gravitational potential energy becomes less negative (increases).