# Hooke's Law

## Hooke's law of elasticity is an approximation that states that the extension of a spring is directly proportional to the load applied to it.

#### Key Points

• Mathematically, Hooke's Law can be written as F=-kx.

• Many materials obey this law as long as the load does not exceed the material's elastic limit.

• The rate or spring constant, k, relates the force to the extension in SI units: N/m or kg/s2.

#### Terms

• The property by virtue of which a material deformed under the load can regain its original dimensions when unloaded

#### Examples

• Many weighing machines, such as scales, use Hooke's Law to measure the mass of an object.

#### Figures

1. ##### Diagram of Hooke's Law

The extension of the spring is linearly proportional to the force.

2. ##### Springs and Hooke's Law

A brief overview of springs, Hooke's Law, and elastic potential energy for algebra-based physics students.

In mechanics (physics), Hooke's law is an approximation of the response of elastic (i.e., springlike) bodies. It states: the extension of a spring is in direct proportion with the load applied to it Figure 2. For instance, in Figure 1 the spring is pulled downwards with either no load, Fp, or twice Fp.

Many materials obey this law of elasticity as long as the load does not exceed the material's elastic limit. Materials for which Hooke's law is a useful approximation are known as linear-elastic or "Hookean" materials. Hookean materials are broadly defined and include springs as well as muscular layers of the heart. In simple terms, Hooke's law says that stress is directly proportional to strain. Mathematically, Hooke's law is stated as:

<equation contenteditable="false">$F=-kx$

where:

• x is the displacement of the spring's end from its equilibrium position (a distance, in SI units: meters);
• F is the restoring force exerted by the spring on that end (in SI units: N or kg·m/s2); and
• k is a constant called the rate or spring constant (in SI units: N/m or kg/s2). When this holds, the behavior is said to be linear. If shown on a graph, the line should show a direct variation.

It's possible for multiple springs to act on the same point. In such a case, Hooke's law can still be applied. As with any other set of forces, the forces of many springs can be combined into one resultant force.

When Hooke's law holds, the behavior is linear; if shown on a graph, the line depicting force as a function of displacement should show a direct variation. There is a negative sign on the right hand side of the equation because the restoring force always acts in the opposite direction of the displacement (for example, when a spring is stretched to the left, it pulls back to the right).

Hooke's law is named after the 17th century British physicist Robert Hooke, and was first stated in 1660 as a Latin anagram, whose solution Hooke published in 1678 as Ut tensio, sic vis, meaning, "As the extension, so the force."

#### Key Term Glossary

approximation
An imprecise solution or result that is adequate for a defined purpose.
##### Appears in these related concepts:
displacement
A vector quantity that denotes distance with a directional component.
##### Appears in these related concepts:
Displacement
The length and direction of a straight line between two objects.
##### Appears in these related concepts:
elastic
Capable of stretching; particularly, capable of stretching so as to return to an original shape or size when force is released.
##### Appears in these related concepts:
elasticity
The property by virtue of which a material deformed under the load can regain its original dimensions when unloaded
##### Appears in these related concepts:
elastic limit
The level of stress at which a solid undergoes a greater change in strain than predicted by Hooke's law; often followed by necking and breaking.
##### Appears in these related concepts:
equation
An assertion that two expressions are equal, expressed by writing the two expressions separated by an equal sign; from which one is to determine a particular quantity.
##### Appears in these related concepts:
equilibrium
The state of a body at rest or in uniform motion, the resultant of all forces on which is zero.
##### Appears in these related concepts:
Equilibrium
A state of rest or balance due to the equal action of opposing forces.
##### Appears in these related concepts:
force
A physical quantity that denotes ability to push, pull, twist or accelerate a body which is measured in a unit dimensioned in mass × distance/time² (ML/T²): SI: newton (N); CGS: dyne (dyn)
##### Appears in these related concepts:
Force
A force is any influence that causes an object to undergo a certain change, either concerning its movement, direction or geometrical construction.
##### Appears in these related concepts:
Law
A concise description, usually in the form of a mathematical equation, used to describe a pattern in nature
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machine
A mechanical or electrical device that performs or assists in the performance of human tasks, whether physical or computational, laborious or for entertainment.
##### Appears in these related concepts:
mass
The quantity of matter which a body contains, irrespective of its bulk or volume. It is one of four fundamental properties of matter. It is measured in kilograms in the SI system of measurement.
##### Appears in these related concepts:
position
A place or location.
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restoring force
If the system is perturbed away from the equilibrium, the restoring force will tend to bring the system back toward equilibrium. The restoring force is a function only of position of the mass or particle. It is always directed back toward the equilibrium position of the system.An example is the action of a spring. An idealized spring exerts a force that is proportional to the amount of deformation of the spring from its equilibrium length, exerted in a direction to oppose the deformation. Pulling the spring to a greater length causes it to exert a force that brings the spring back toward its equilibrium length. The amount of force can be determined by multiplying the spring constant of the spring by the amount of stretch.
##### Appears in these related concepts:
Restoring force
A variable force that gives rise to an equilibrium in a physical system. If the system is perturbed away from the equilibrium, the restoring force will tend to bring the system back toward equilibrium. The restoring force is a function only of position of the mass or particle. It is always directed back toward the equilibrium position of the system
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resultant
A vector that is the vector sum of multiple vectors
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SI units
International System of Units (abbreviated SI from French: Le Système international d'unités). It is the modern form of the metric system.
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strain
The amount by which a material deforms under stress or force, given as a ratio of the deformation to the initial dimension of the material and typically symbolized by ε is termed the engineering strain. The true strain is defined as the natural logarithm of the ratio of the final dimension to the initial dimension.
##### Appears in these related concepts:
stress
The internal distribution of force per unit area (pressure) within a body reacting to applied forces which causes strain or deformation and is typically symbolized by σ.