Springs and Hooke's Law

Hooke's law

The force exerted by a spring on an object attached to it is proportional to the displacement from equilibrium:

F = -k·x

where:

• x is displacement from equilibrium (positive = stretched, negative = compressed).
• k is the spring constant (always positive, units N/m).
• The negative sign means the force opposes displacement (restoring).

Stretch the spring → it pulls back. Compress it → it pushes back. Always toward equilibrium.

Spring constant — what determines k?

Factor	Effect on k
Wire material (Young's modulus E)	Higher E → stiffer (more rigid wire)
Wire diameter d	k ∝ d⁴ (very strong dependence)
Coil radius R	k ∝ 1/R³ (smaller coils → stiffer)
Number of coils N	k ∝ 1/N (fewer coils → stiffer)

Combined formula for a coil spring: k = G·d⁴/(8·D³·N) where G is shear modulus, D is mean coil diameter, N is number of coils.

Elastic potential energy

The work to stretch a spring from 0 to x:

U(x) = ∫₀ˣ k·x' dx' = ½·k·x²

This is the potential energy stored in the spring. It's quadratic in displacement — a parabola.

For x = ±A, U = ½kA² (max). At x = 0, U = 0. Total mechanical energy in undamped SHM:

E = ½kA² (constant)
KE + PE = E (oscillates between forms)

Series and parallel combinations

Configuration	k_eff	Notes
Single spring, k	k	—
Two in series, k₁ and k₂	1/(1/k₁ + 1/k₂)	Same force, displacements add
Two in parallel, k₁ and k₂	k₁ + k₂	Same displacement, forces add
Two equal springs, k each, series	k/2	Half stiffness
Two equal springs, k each, parallel	2k	Double stiffness
n in series	k/n (if all equal)	Long springs are softer
n in parallel	n·k	Multiple springs share load

Mass-spring oscillation

For a mass m on a spring of constant k, the equation of motion:

m·ẍ = -k·x  →  ẍ = -(k/m)·x = -ω²·x

This is SHM with ω = √(k/m). Period:

T = 2π · √(m/k)

Doubling mass: T increases by √2 (~1.41×). Quadrupling stiffness: T halves.

Real springs — beyond Hooke

Hooke's law works only in the elastic region. For larger displacements:

Region	Behavior
Elastic (small x)	F = -kx (Hooke); spring fully recovers
Yield region	Permanent deformation begins (plastic flow)
Plastic deformation	Returns to "set" shape with offset
Breaking point	Spring snaps (fractures)
Compressed beyond limit	Coils touch (further compression impossible)

Spring designers build in safety margins — operating range is 30-50% of yield.

JavaScript — spring problems

// Force from spring at displacement x
function springForce(k, x) {
  return -k * x;
}

// Elastic PE
function elasticPE(k, x) {
  return 0.5 * k * x * x;
}

// Period of mass-spring oscillator
function springPeriod(m, k) {
  return 2 * Math.PI * Math.sqrt(m / k);
}

// Series and parallel combinations
function seriesK(k_array) {
  return 1 / k_array.reduce((s, k) => s + 1/k, 0);
}

function parallelK(k_array) {
  return k_array.reduce((s, k) => s + k, 0);
}

console.log(`2 springs of 100 N/m series: ${seriesK([100, 100])} N/m`);  // 50
console.log(`2 springs of 100 N/m parallel: ${parallelK([100, 100])} N/m`); // 200

// Spring launches a ball — find launch velocity
function springLaunch(k, x_compressed, mass) {
  // ½kx² → ½mv² (energy conservation)
  const PE = elasticPE(k, x_compressed);
  return Math.sqrt(2 * PE / mass);
}

// 100 N/m spring compressed 10 cm, ball mass 0.05 kg
const launch_v = springLaunch(100, 0.1, 0.05);
console.log(`Launch velocity: ${launch_v.toFixed(2)} m/s`); // 4.47 m/s

// Range from spring launch (horizontal projectile from height h)
function rangeFromSpring(k, x_compressed, mass, height_above_ground) {
  const v = springLaunch(k, x_compressed, mass);
  const t_fall = Math.sqrt(2 * height_above_ground / 9.81);
  return v * t_fall;
}

console.log(`Range (1 m up): ${rangeFromSpring(100, 0.1, 0.05, 1).toFixed(2)} m`); // 2.02 m

// Pendulum-like time from spring constant equivalent
// (Spring energy of mass-on-string acts like restoring constant k = mg/L)
function springConstantOfPendulum(m, L, g = 9.81) {
  return m * g / L;  // effective k for small oscillations
}

console.log(`1m pendulum equivalent k: ${springConstantOfPendulum(1, 1).toFixed(2)} N/m`);  // 9.81

Where Hooke's law shows up

• Mechanical engineering. Vehicle suspension, vibration isolation, watch escapements, valve springs, locks.
• Material science. Young's modulus, bulk modulus, shear modulus — all derive from Hooke's law applied to bulk materials.
• Molecular physics. Bond stretching/compression near equilibrium follows F = -kx; bond force constants from IR spectroscopy.
• Acoustics. Air pressure waves — local pressure deviation acts like spring restoring force; sound waves are Hooke-like oscillations.
• Microelectromechanical (MEMS). Silicon springs in accelerometers (smartphone gyros), pressure sensors, micromirrors.
• Architecture. Tuned mass dampers (large pendulums) in skyscrapers act as springs; building structural elements treated as composite springs.
• Toys and sports. Trampolines, pogo sticks, archery bows, gymnastics floor springs.

Common mistakes

• Forgetting the negative sign. F = -kx, not +kx. Force is restoring (toward equilibrium).
• Applying Hooke beyond elastic limit. Overstretching a spring permanently deforms it; F = -kx no longer holds.
• Confusing series and parallel. Springs in series → softer (1/k_eff = sum of 1/k). Springs in parallel → stiffer (k_eff = sum of k). Opposite of intuition for some.
• Wrong PE formula. Spring PE is ½kx², NOT kx². The factor of ½ comes from integrating force over distance.
• Mixing pendulum and spring formulas. Pendulum: T = 2π√(L/g). Spring: T = 2π√(m/k). Both look like SHM but have different physical sources.
• Treating real springs as ideal. Real springs have mass, internal damping, non-linearity, and fatigue. Hooke's law is the linear approximation.