Hang a mass on a spring and pull it down. Hooke's law says the spring pulls back with a force proportional to the stretch; friction, if there is any, resists in proportion to the speed; and anything else pushing on the mass is the forcing. Newton's law adds these up into a second-order linear equation with constant coefficients — the mass, the friction and the stiffness — in which each of the three numbers means something you could point to.

A car on its springs and shock absorbers. The springs push back, the dampers resist motion, and the road does the forcing. Every term of the equation is a part under the car.

Hooke's law $F=-kx$, drawn for $k=2$ with the displacement across and the force up. Stretched, the spring pulls back; compressed, it pushes out: the force always points towards equilibrium.

For a mass $m$ on a spring of stiffness $k$, with friction coefficient $c\ge0$ and applied force $F(t)$, the displacement $x$ from equilibrium satisfies . Measuring from the equilibrium position removes gravity, because the spring's stretch there balances the weight. The natural frequency is how fast the system oscillates when left alone without friction.

• Stiffer springs and lighter masses oscillate faster: $\omega_{0}=\sqrt{k/m}$.

Mass times acceleration equals the sum of the forces: the spring pulls back, friction resists motion, and the applied force pushes.

A resistor, an inductor and a capacitor in series obey the spring equation for the charge, term for term.

Great work! You now know how to:

• build the spring equation from Newton’s and Hooke’s laws
• say what each coefficient means and compute the natural frequency
• explain why gravity drops out when measuring from equilibrium
• translate between a spring and a series circuit