Hooke's Law Calculator

About Hooke's Law Calculator

What Is Hooke's Law?

Hooke's Law states that the restoring force of an elastic object is directly proportional to its displacement from equilibrium, provided the displacement stays within the elastic limit. The formula is F = kx (magnitude) or F = -kx (vector, where the negative sign indicates the force opposes the displacement). Robert Hooke first published the relationship in 1676 as the anagram "ceiiinosssttuv" and revealed the Latin solution "Ut tensio, sic vis" ("as the extension, so the force") in 1678 in his Lectures De Potentia Restitutiva, per the Encyclopaedia Britannica.

Solve For	Formula	Example
Force	F = k x x	200 N/m x 0.15 m = 30 N
Spring constant	k = F / x	30 N / 0.15 m = 200 N/m
Displacement	x = F / k	30 N / 200 N/m = 0.15 m

Worked example: a spring with k = 500 N/m is stretched 8 cm (0.08 m). Force = 500 x 0.08 = 40 N. Energy stored = 1/2 x 500 x 0.08² = 1.6 J. If released, this energy converts to kinetic energy, which is how spring-loaded mechanisms work.

Elastic Potential Energy

A deformed spring stores potential energy: PE = 1/2 kx². Like kinetic energy's velocity-squared term, the squared displacement means doubling the stretch quadruples the stored energy.

Displacement (cm)	Force (N) at k=300 N/m	Energy Stored (J)	Relative Energy
1	3	0.015	1x
2	6	0.06	4x
5	15	0.375	25x
10	30	1.5	100x
20	60	6.0	400x

Typical Spring Constants

Spring / System	Spring Constant (N/m)	Context
Slinky toy	~1	Very soft, stretches under own weight
Ballpoint pen spring	~100	Light click action
Screen door spring	~200 - 500	Moderate closing force
Trampoline spring (single)	~3,000 - 5,000	Must support body weight with modest stretch
Garage door spring	~10,000 - 20,000	Counterbalances heavy door
Car suspension spring	~20,000 - 40,000	Supports 1/4 of vehicle weight
Truck suspension spring	~100,000 - 200,000	Heavy load support
Building earthquake isolator	~1,000,000+	Absorbs seismic energy

The Elastic Limit and Beyond

Hooke's Law only applies within the elastic region of a material - the range where deformation is reversible. Beyond the elastic limit (yield point), permanent deformation occurs and the linear relationship breaks down.

Region	Behaviour	What Happens to the Spring
Elastic (Hooke's Law applies)	F proportional to x, fully reversible	Returns to original shape when released
Yield point	Transition from elastic to plastic	Begins to permanently stretch
Plastic deformation	F and x no longer proportional	Stays deformed when released
Fracture point	Material breaks	Spring snaps

Springs in Series and Parallel

When springs are combined, the effective spring constant changes.

Configuration	Effective k	Example (k₁=200, k₂=300)	Effect
Parallel (side by side)	k_eff = k₁ + k₂	200 + 300 = 500 N/m	Stiffer - both resist together
Series (end to end)	1/k_eff = 1/k₁ + 1/k₂	1/200 + 1/300 = 1/120 → 120 N/m	Softer - each stretches independently

This is the opposite of electrical resistors: springs in parallel add (like resistors in series), and springs in series use the reciprocal formula (like resistors in parallel).

Applications of Hooke's Law

Application	How Hooke's Law Applies
Spring scales (weighing)	Extension proportional to weight gives a linear scale
Vehicle suspension	Springs absorb bumps; k determines ride comfort vs handling
Mechanical watches	Hairspring oscillation period depends on k and mass
Bungee cords	Force increases with stretch, decelerating the jumper gradually
Atomic bonds	Small vibrations around equilibrium are approximately Hookean
Seismometers	Spring-mass system detects ground vibrations

Simple Harmonic Motion and Oscillation

A mass attached to a Hookean spring oscillates in simple harmonic motion (SHM). The period depends only on the mass and spring constant: T = 2π √(m/k). Stiffer springs (higher k) oscillate faster; heavier masses (higher m) oscillate slower. The frequency f = 1/T gives the oscillations per second in hertz.

Worked example: a 0.5 kg mass on a spring with k = 200 N/m has period T = 2π √(0.5/200) = 2π × 0.05 = 0.314 s, so f ≈ 3.18 Hz. This is the principle behind pendulum-free mechanical watch balance wheels, quartz crystal oscillators (at a much higher frequency), and car suspension tuning. For projectile and momentum problems after a spring launches an object, the momentum calculator handles p = mv conversions.

Young's Modulus: Hooke's Law for Materials

Hooke's Law generalises from springs to any elastic material via Young's modulus (E), which relates stress (force per area) to strain (fractional length change): σ = Eε. Each material has a characteristic E measured in pascals (Pa) or gigapascals (GPa). Values below are from the Engineering ToolBox and NIST materials data.

Material	Young's Modulus (GPa)	Typical Use
Rubber	0.01 - 0.1	Seals, shock absorbers
Wood (oak, along grain)	11	Construction, furniture
Bone (human, cortical)	14	Skeleton
Concrete	30	Buildings, foundations
Aluminium	69	Aircraft, cans
Brass	100 - 125	Fittings, musical instruments
Steel	200	Structural beams, springs
Tungsten	411	Light bulb filaments
Diamond	1,050 - 1,200	Hardest natural material

The spring constant of a helical spring itself depends on the wire's shear modulus (G), coil diameter, wire diameter, and number of active coils: k = Gd⁴ / (8D³N), where d is wire diameter, D is mean coil diameter, and N is the number of active coils. Halving the coil diameter multiplies k by eight; this is why a compact spring is much stiffer than a loose one of the same wire.

What Is the Elastic Limit?

The elastic limit is the maximum stress a material can take before permanent deformation begins. For mild steel, the proportional limit is around 210 MPa and the yield stress around 250 MPa, per ASM International materials data. Stretch a paperclip slightly and it springs back; bend it sharply and it stays bent because you exceeded the yield stress. Structural engineers design with safety factors of 1.5 to 3 below the yield stress so everyday loads stay comfortably within the Hookean region.

Region	Behaviour	What Happens to the Spring
Elastic (Hooke's Law applies)	F proportional to x, fully reversible	Returns to original shape when released
Yield point	Transition from elastic to plastic	Begins to permanently stretch
Plastic deformation	F and x no longer proportional	Stays deformed when released
Fracture point	Material breaks	Spring snaps

Metal fatigue is a separate failure mode: a spring cycled repeatedly well below its yield stress can still fail after millions of cycles because microscopic cracks grow on each load cycle. Automotive valve springs, for example, are rated for roughly 200 million cycles (about 150,000 miles of driving) before replacement becomes prudent.

Common Mistakes and Edge Cases

• Mixing units. Using centimetres for displacement with N/m for the spring constant gives results 100x too low. Always convert to metres before applying F = kx in SI.
• Forgetting the sign. In vector form, F = -kx; the force opposes displacement. Many exam problems expect the sign explicitly; if a question asks for the "restoring force" as a vector, include the negative.
• Assuming constant k over any range. Real springs deviate from linearity near the coil-binding point (fully compressed) or near yield (fully stretched). The listed k values apply only in the middle of the operating range.
• Doubling stretch, doubling energy. Energy scales with x², not x. Doubling the stretch quadruples the stored energy - this is why a slightly overstretched trampoline spring stores far more energy than intuition suggests.
• Ignoring spring mass. The T = 2π √(m/k) formula assumes a massless spring. For a real spring with mass m_s, replace m with m + m_s/3 for more accurate oscillation periods.
• Using static k for dynamic loads. Sudden impact loads can momentarily double the effective force compared with gently placing the same weight on a spring - a classic result from energy conservation. Design for impact, not just weight.
• Treating rubber bands as Hookean. Rubber is only approximately linear over a narrow range. At large stretches the stress-strain curve becomes S-shaped (Mullins effect), so spring-constant calculations for rubber bands are rough estimates at best.

Real-World Measurements and Benchmarks

A standard mousetrap spring has k ≈ 100-200 N/m with about 0.05 m of cocked displacement, storing around 0.25 J - enough kinetic energy at release to accelerate the bar at roughly 2,400 m/s². An Olympic archery bow at full draw (0.7 m, draw weight 220 N) stores about 77 J; this same energy, transferred to a 20 g arrow, gives a muzzle velocity of around 88 m/s, matching published arrow speeds. A typical car suspension spring compresses roughly 5 cm when a 75 kg adult sits in the driver's seat, implying k ≈ 15,000 N/m per corner - consistent with the 10,000-50,000 N/m range in the reference table above. For further kinetic-to-potential energy conversions, see the kinetic energy calculator; for pure force problems, the force calculator applies F = ma with full unit support.

Sources

• Encyclopaedia Britannica - Hooke's Law
• OpenStax College Physics - Hooke's Law, Stress and Strain
• Engineering ToolBox - Young's Modulus of Materials
• NIST - SI Units for Mechanics (Force, Energy, Pressure)
• ASM International - Materials Properties Handbook
• Feynman Lectures - The Harmonic Oscillator