Hooke's Law Calculator

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Force F

Spring Constant k

Displacement x

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What Is Hooke's Law?

Hooke's Law, formulated by Robert Hooke in 1678, describes the elastic behaviour of springs and many solid materials. Within the elastic limit, the restoring force is proportional to the displacement:

F = k · x

F = force (N), k = spring constant (N/m), x = displacement (m)

The formal statement uses F = −kx (the negative sign indicates the restoring direction), but engineers typically work with magnitudes. The law is the foundation of vibration analysis, structural engineering, and simple harmonic motion.

Spring Constant (k) Explained

The spring constant k quantifies stiffness — how much force is needed per unit of deformation. Higher k = stiffer spring.

Example	k (N/m)
Pen click spring	~148
Archery bow	~200
Mattress spring	~4,000
Pogo stick	~10,000
Car suspension (per wheel)	~20,000
Engine valve spring	~50,000

Elastic Potential Energy

Work done to compress or stretch a spring is stored as elastic potential energy. Because the force increases linearly with displacement, the energy is:

E = ½ · k · x²

This energy can be converted into kinetic energy (bungee cord, spring launcher) or do useful work (mechanical watch mainspring). The ½ factor arises because work = area under the force-displacement triangle.

Series vs Parallel Springs

Series (k_eq is smaller)

1/k_eq = 1/k₁ + 1/k₂ + ...

Springs share force; total extension = sum of extensions. Used when softer response is needed.

Parallel (k_eq is larger)

k_eq = k₁ + k₂ + ...

Springs share load; total extension is the same for each. Used to increase stiffness (car leaf springs).

Beyond the Elastic Limit

Hooke's Law is valid only up to the proportional limit. Beyond this:

Elastic limit: material still returns to original shape, but the F-x relationship is no longer linear.
Yield point: permanent (plastic) deformation begins.
Ultimate strength: maximum stress before fracture.

For most steel springs, the elastic limit is at roughly 60–70% of tensile strength.

Engineering Applications

Hooke's Law underpins a wide range of engineering systems:

Vehicle Suspension

Coil springs with k ≈ 15,000–25,000 N/m absorb road shocks and maintain tire contact.

Vibration Isolation

Machine mounts use known k to tune natural frequency away from operating frequencies.

Force Measurement

Spring scales and force gauges exploit F = kx; displacement indicates force magnitude.

MEMS Sensors

Micro-scale springs in accelerometers and gyroscopes follow Hooke's Law at microscopic deflections.

Worked Examples

Example 1 — Car Suspension

Given: k = 20,000 N/m, x = 0.05 m

F = k × x = 20000 × 0.05

F = 1,000 N

E = ½ × 20000 × 0.05² = 25 J

Example 2 — Find Spring Constant

Given: F = 50 N, x = 0.02 m

k = F / x = 50 / 0.02

k = 2,500 N/m

Example 3 — Series Combination

k₁ = 100 N/m, k₂ = 200 N/m

1/k_eq = 1/100 + 1/200 = 0.015

k_eq = 66.7 N/m

Example 4 — Bow String Energy

k = 200 N/m, x = 0.7 m

E = ½ × 200 × 0.7²

E = 49 J

Frequently Asked Questions

Hooke's Law states that the force required to extend or compress a spring is directly proportional to the displacement: F = kx, where F is force in newtons, k is the spring constant in N/m, and x is displacement in metres. It holds as long as the material stays within its elastic limit.

The spring constant k (also called stiffness coefficient) measures how stiff a spring is. A higher k means a stiffer spring requiring more force per unit of displacement. It is measured in N/m (newtons per metre). Values range from ~0.1 N/m for soft springs to millions of N/m for industrial springs.

The negative sign in F = −kx indicates that the spring force acts opposite to the displacement — it is a restoring force. When you stretch a spring (positive displacement), the force pulls back (negative direction). This sign convention is important for oscillation and SHM analysis. This calculator uses the magnitude form F = kx for simplicity.

The elastic limit is the maximum stress or deformation a material can undergo and still return to its original shape when the force is removed. Beyond the elastic limit, permanent (plastic) deformation occurs and Hooke's Law no longer applies. For steel springs, the elastic limit is typically 60–70% of the yield strength.

In series, the equivalent spring constant is 1/k_eq = 1/k₁ + 1/k₂ + ... — the total stiffness is less than any individual spring, like weak links in a chain. In parallel, k_eq = k₁ + k₂ + ... — stiffness adds up, making the combination stiffer. A car's suspension uses parallel springs for load sharing.

The elastic potential energy stored in a spring is E = ½kx², where k is the spring constant and x is the displacement from equilibrium. This energy is fully recoverable within the elastic limit. For example, a spring with k = 200 N/m compressed by 0.1 m stores E = ½ × 200 × 0.01 = 1 J.

Hooke's Law is a linear approximation that works well for metals (steel, brass, aluminium) within their elastic range. It does not apply to rubber (which is non-linear), biological tissues, or materials beyond their elastic limit. Many engineering materials follow Hooke's Law up to their proportional limit, which is below the elastic limit.

The spring constant k is measured in N/m (newtons per metre) in SI units. Equivalent units include N/cm, N/mm, and lbf/in. Conversion: 1 N/cm = 100 N/m; 1 N/mm = 1000 N/m; 1 lbf/in ≈ 175.127 N/m. In the CGS system, k would be in dyn/cm = 0.001 N/m.

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