Elastic Potential Energy Reference Cheat Sheet

Key Facts

• Hooke’s law for an ideal spring is $F_s = -kx$, where $F_s$ is spring force, $k$ is spring constant, and $x$ is displacement from equilibrium.
• The magnitude of the spring force is $|F_s| = k|x|$, so a stiffer spring or larger stretch creates a larger force.
• Elastic potential energy stored in an ideal spring is $U_s = \frac{1}{2}kx^2$.
• The spring constant has units of newtons per meter, written as $\text{N}/\text{m}$.
• Displacement in spring equations must be measured from the equilibrium position, not from the spring’s unstretched length unless that is the equilibrium point.
• Elastic potential energy is always nonnegative because $x^2$ is nonnegative in $U_s = \frac{1}{2}kx^2$.
• If no nonconservative work is done, mechanical energy can be conserved using $K_i + U_{g,i} + U_{s,i} = K_f + U_{g,f} + U_{s,f}$.
• The work done by an external force to slowly stretch an ideal spring from $0$ to $x$ is $W = \frac{1}{2}kx^2$.

Vocabulary

Elastic potential energy

Energy stored in an object when it is stretched or compressed, often calculated for a spring with $U_s = \frac{1}{2}kx^2$.

Hooke’s law

The rule that an ideal spring exerts a restoring force given by $F_s = -kx$.

Spring constant

A measure of spring stiffness, represented by $k$, with units $\text{N}/\text{m}$.

Displacement

The signed distance $x$ that a spring is stretched or compressed from its equilibrium position.

Restoring force

A force that acts opposite the displacement and tends to return a spring to equilibrium.

Mechanical energy

The total energy from motion and position, often written as $E = K + U_g + U_s$ in spring problems.