50
N
0.1
m
100
mm
500
N/m
2.5
J
2.5
J
2.5
J
50
N
0.1
m
100
mm
500
N/m
2.5
J
2.5
J
2.5
J
The Hooke’s Law Calculator applies the fundamental law of elasticity to compute the restoring force, displacement, or spring constant of a spring, along with the associated elastic potential energy and work done. Hooke’s law is one of the cornerstones of classical mechanics and materials science.
Formulated by Robert Hooke in 1660, Hooke’s law states that the force needed to extend or compress a spring by some distance is proportional to that distance: $$F = -kx$$ Here F is the restoring force exerted by the spring, k is the spring constant (stiffness), and x is the displacement from the equilibrium position. The negative sign indicates the force opposes the displacement, always pushing or pulling the object back toward equilibrium.
This elegant linear relationship holds for virtually all elastic materials at sufficiently small deformations. Steel beams, rubber bands, bones, and even the crust of the Earth all approximately obey Hooke’s law within their elastic limits. Beyond the elastic limit, materials undergo permanent deformation and the linear relationship breaks down.
The elastic potential energy stored in a spring displaced by x from equilibrium is $$PE = \frac{1}{2}kx^2$$ This energy represents the work done on the spring by an external agent to deform it from its natural length. When released, this stored energy converts to kinetic energy, driving the oscillatory motion characteristic of spring systems.
The work done by the spring (the restoring force) as it returns to equilibrium is $$W_{spring} = -\frac{1}{2}kx^2$$ which is negative when the spring is being deformed (the spring opposes the motion) and positive when it returns to its natural length. The work done on the spring by an external force is the negative of this: $$W_{external} = \frac{1}{2}kx^2$$
This calculator operates in three modes, solving for whichever quantity you do not know. It provides the complete energy picture alongside the force-displacement relationship, making it ideal for physics problem-solving, engineering analysis, and educational demonstrations of elastic behavior.
The calculator solves Hooke’s law in three configurations:
Mode 1: Solve for Force
$$F = -kx$$
Given the spring constant and displacement, compute the restoring force.
Mode 2: Solve for Displacement
$$x = \frac{F}{k}$$
Given the applied force and spring constant, compute how far the spring stretches or compresses.
Mode 3: Solve for Spring Constant
$$k = \frac{F}{x}$$
Given the force and displacement, determine the spring stiffness.
In all modes, the calculator also computes:
$$PE = \frac{1}{2}kx^2, \quad W_{spring} = -\frac{1}{2}kx^2, \quad W_{external} = \frac{1}{2}kx^2$$
The restoring force is negative when displacement is positive (or vice versa), confirming the spring pulls/pushes back toward equilibrium. The elastic potential energy is always positive or zero. Work done by the spring is negative during deformation (spring resists) and represents energy stored. Work done on the spring equals the potential energy gained. These energy values are path-independent for ideal springs.
Inputs
Results
An 800 N/m spring stretched 3 cm exerts a restoring force of 24 N and stores 0.36 J of elastic potential energy. An external agent must do 0.36 J of work to achieve this deformation.
Inputs
Results
A 75 N force applied to a 1500 N/m spring produces 5 cm of displacement. The potential energy stored is 1.875 J.
Hooke’s law states that the restoring force of an elastic body is directly proportional to the displacement from its natural shape: F = −kx. It was first stated by Robert Hooke in 1660 as a Latin anagram and published in 1678. It applies to springs, elastic beams, rubber bands, and any material deformed within its elastic limit.
Hooke’s law fails when the material is deformed beyond its elastic limit (yield point). Beyond this, the material undergoes plastic (permanent) deformation and the force-displacement relationship becomes nonlinear. Rubber at large extensions, metals past yield, and brittle materials near fracture all violate Hooke’s law.
The negative sign indicates the restoring nature of the force: it always acts in the direction opposite to the displacement. If you stretch a spring to the right (+x), the spring pulls back to the left (−F). This opposing character is what makes the spring return to equilibrium and enables oscillatory motion.
Elastic potential energy is the energy stored in a spring (or any elastic body) due to its deformation. It equals PE = ½kx² and represents the work required to deform the spring from its natural length. This energy is recovered when the spring returns to its natural shape, converting to kinetic energy or doing work on other objects.
Yes. Hooke’s law applies equally to compression (negative x) and extension (positive x). The formula F = −kx works in both directions. For compression, x is negative, so the force is positive (pushing outward). The energy formula PE = ½kx² gives the same positive energy for equal compression and extension.
No. While springs are the iconic example, Hooke’s law describes the elastic behavior of any material near equilibrium. It applies to bending beams, torsion wires, stretched wires, inflated balloons, and even atomic bonds. Any restoring force that is proportional to displacement follows Hooke’s law, which is why it appears across all branches of physics and engineering.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
How helpful was this calculator?
Be the first to rate!
Simple Harmonic Motion Calculator
Oscillations & Waves Calculators
Spring Calculator
Oscillations & Waves Calculators
Spring Constant Calculator
Oscillations & Waves Calculators
Pendulum Calculator
Oscillations & Waves Calculators
Pendulum Period Calculator
Oscillations & Waves Calculators
Pendulum Frequency Calculator
Oscillations & Waves Calculators
