Hooke's law Definition and 38 Discussions

Hooke's law is a law of physics that states that the force (F) needed to extend or compress a spring by some distance (x) scales linearly with respect to that distance—that is, Fs = kx, where k is a constant factor characteristic of the spring (i.e., its stiffness), and x is small compared to the total possible deformation of the spring. The law is named after 17th-century British physicist Robert Hooke. He first stated the law in 1676 as a Latin anagram. He published the solution of his anagram in 1678 as: ut tensio, sic vis ("as the extension, so the force" or "the extension is proportional to the force"). Hooke states in the 1678 work that he was aware of the law since 1660.
Hooke's equation holds (to some extent) in many other situations where an elastic body is deformed, such as wind blowing on a tall building, and a musician plucking a string of a guitar. An elastic body or material for which this equation can be assumed is said to be linear-elastic or Hookean.
Hooke's law is only a first-order linear approximation to the real response of springs and other elastic bodies to applied forces. It must eventually fail once the forces exceed some limit, since no material can be compressed beyond a certain minimum size, or stretched beyond a maximum size, without some permanent deformation or change of state. Many materials will noticeably deviate from Hooke's law well before those elastic limits are reached.
On the other hand, Hooke's law is an accurate approximation for most solid bodies, as long as the forces and deformations are small enough. For this reason, Hooke's law is extensively used in all branches of science and engineering, and is the foundation of many disciplines such as seismology, molecular mechanics and acoustics. It is also the fundamental principle behind the spring scale, the manometer, the galvanometer, and the balance wheel of the mechanical clock.
The modern theory of elasticity generalizes Hooke's law to say that the strain (deformation) of an elastic object or material is proportional to the stress applied to it. However, since general stresses and strains may have multiple independent components, the "proportionality factor" may no longer be just a single real number, but rather a linear map (a tensor) that can be represented by a matrix of real numbers.
In this general form, Hooke's law makes it possible to deduce the relation between strain and stress for complex objects in terms of intrinsic properties of the materials it is made of. For example, one can deduce that a homogeneous rod with uniform cross section will behave like a simple spring when stretched, with a stiffness k directly proportional to its cross-section area and inversely proportional to its length.

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  1. nomadreid

    Calculate the distance a projectile travels using a rough approximation (Hooke's Law) for a slingshot

    This concerns an elementary experiment that I (a teacher) have done with several secondary school classes, up until now with success. However, I gave the same instructions to a homeschooling student (in another country , so I couldn’t actually directly oversee the experiment), and the...
  2. A

    Fluid mechanics - Additional liquid capacity due to compression

    A cylindrical tube (diameter = D, width = L) is completely filled with a liquid (density = ρ). A pump pressurizes the system with a pressure P. Consequently, 1) the solid tube is compressed and deformed according to Hooke's law (σ = ε.E), and 2) the liquid is compressed and deformed, following...
  3. Amaterasu21

    B Why is Young's modulus constant below the limit of proportionality?

    Hi all, I'm a little confused about something. Force-extension graphs and stress-strain graphs are always both straight lines up until the limit of proportionality, implying both the spring constant and the Young modulus are constant up until then. For a force-extension graph, Hooke's Law...
  4. B

    Formula for the energy of elastic deformation

    In every book I checked, the energy (per unit mass) of elastic deformation is derived as follows: ## \int \sigma_1 d \epsilon_1 = \frac{\sigma_1 \epsilon_1}{2} ## and then, authors (e.g. Timoshenko & Goodier) sum up such terms and substitute ##\epsilon ## from generalised Hooke's law i.e. ##...
  5. O

    What's the importance of a spring force constant of 1634N/m?

    Hi I'm new here and I've checked everywhere on google but I can't seem to find a website that'll tell me the spring force constant of items. Also what things would be in the range of a spring force constant of 163.427 N/m/
  6. Jeviah

    Spring characteristics when loaded with a mass

    Homework Statement If a spring is loaded with a mass, the spring being completely vertical and the mass hanging below it there would be an extension in the spring relative to the mass applied/force applied according to hookes law. If the same spring was not vertical but coiled (similar to a...
  7. D

    How do you know when k in Hooke's law is positive or negativ

    Homework Statement A 7.2-kg mass is hanging from the ceiling of an elevator by a spring of spring constant 150N/m whose unstretched length is 80 cm. What is the overall length of the spring when the elevator: (a) starts moving upward with acceleration 0.95m/s2 ; (b) moves upward at a steady...
  8. navneet9431

    Is it possible to apply energy conservation here?

    Homework Statement Homework Equations Kinetic Energy =1/2*m*v^2 Spring Potential Energy = 1/2*k*x^2 Gravitational Potential Energy = m*g*h The Attempt at a Solution I am thinking to solve this problem using energy conservation but I feel that it is not possible to apply energy conservation...
  9. astroman707

    Derive the formula for the frequency of a spring

    Homework Statement Two masses m1 and m2 are joined by a spring of spring constant k. Show that the frequency of vibration of these masses along the line connecting them is ω = √[ k(m1 + m2) / (m1*m2) ] (Hint: Center of mass remains at rest.) Homework Equations f = w/2π w = √(k/m) F = -kx a = -...
  10. jxj

    Help with proportionality and literal equations

    Homework Statement Its a series of problems essentially basically asking questions about solving proportionality . For example "Hooke's Law of a spring can be described by the equation F = -kx, where F is the force exerted by a spring, K us the spring constant, and X is the amount of distance...
  11. A

    Is energy profile of a spring asymmetric with +x and -x?

    Homework Statement [/B] The problem is with part b(ii). This is an A level question so only elementary concepts are to be used. Homework Equations $$E = \frac{1}{2}kx^2 $$ The Attempt at a Solution EPE should be max at 0, 0.4, 0.8 since E is a function of $x^2$. However, mark scheme only...
  12. thebosonbreaker

    Resolving forces involving an elastic string

    Homework Statement A light elastic string of natural length 0.3m has one end fixed to a point on a ceiling. To the other end of the string is attached a particle of mass M. When the particle is hanging in equilibrium, the length of the string is 0.4m. (a) Determine, in terms of M and g (take g...
  13. C

    Mass attached by two springs

    1. Homework Statement The following problem is an example from the book ' Berkely - Waves by Frank S. Crawford Jr '. Mass 'M' slides on a frictionless surface. It is connected to rigid walls by means of two identical springs, each of which has zero mass, spring constant 'K' and relaxed length...
  14. H

    Mechanics - Hooke's law and energy conservation

    Homework Statement One end of a light elastic string of stiffness mg/l and natural length l is attached to a point O. A small bead of mass m is fixed to the free end of the string. The bead is held at O and then released so that it will fall vertically. In terms of find the greatest depth to...
  15. A

    Yield Point vs Elastic Limit

    I don't understand the difference between the elastic limit and the yield point. I understand that if you stretch a material within the elastic limit, then the material should return to its normal shape. However, the yield point is described as the point at which a permanent increase in length...
  16. A

    Elastic potential energy problem

    Homework Statement A 1.00kg mass and 2.00kg mass are set gently on a platform mounted on an ideal spring of force constant 40.0 N/m. The 2.00 kg mass is suddenly removed. How high above its starting position does the 1.00 kg mass reach? Related to it... An 87 g box is attached to a spring with...
  17. C

    Confused about Hooke's law when analyzed at different times

    Homework Statement A.[/B] Suppose I have a block of mass 'M' that is attached to a wall via spring of coefficient 'k' , the spring has rest length Xo . Suppose I look at the problem at some time 't' such that the spring is being compressed and the block moves left ( moving towards x = 0 ) ...
  18. J

    Finding the Spring Constant

    Homework Statement Force Extension in (m) 1.0 0.014. 1.5 0.032. 2.0 0.053. 2.5 0.071. 3.0 0.090. 3.5 0.110. 4.0 0.130. 4.5 0.148. 5.0 0.166...
  19. S

    Approximating a spring constant for an air leg

    Hi all, In short: For an air leg or air spring, there is a method using a Taylor approximation to find the spring constant for very small displacements, but I can't seem to figure out how it works. I've learned that air legs don't follow Hooke's law very much at all, except for when the...
  20. L

    Spring within a spring

    So I am doing tensile testing on an elastic electrical lead for biomedical purposes. The lead is encapsulated in an elastic tubing. Now the lead acts like a weak spring itself (coiled wire). I'm curious, if there are two springs with different k constants "within" each-other (one inside the...
  21. Elvis 123456789

    Determining angular frequency and amplitude for SHM

    Homework Statement A mass "m" is attached to a spring of constant "k" and is observed to have an amplitude "A" speed of "v0" as it passes through the origin. a) What is the angular frequency of the motion in terms of "A" and "v0"? b) Suppose the system is adjusted so that the mass has speed...
  22. I

    Help with spring stiffness calculation (k)

    Hi I was given the following problem in my coursework: Homework Statement "A spring is initially compressed by 50mm when a steel ball of mass 2kg is released from just being in contact with the uncompressed spring. Determine the spring stiffness (k) of the spring." Homework Equations F = mg...
  23. K

    Updating the suspension of a car

    Homework Statement A car driver updates the springs of a car by replacing the old springs with stiffer ones. The old springs give an amount of 8 cm when under the car, and their length when not under the car is 29 cm. The spring constant of the new springs is 30 % greater than that of the old...
  24. B

    Spring Force and mass

    Homework Statement A 5.3kg mass hangs vertically from a spring with spring constant 720N/m. The mass is lifted upward and released. Calculate the force and acceleration the mass when the spring is compressed by 0.36m. Note: I already solved for acceleration and I got the correct answer-...
  25. sushichan

    [Mechanics] Tension in bungee jumping

    Homework Statement A bungee jumper of mass 60kg jumps from a bridge 24 m above the surface of the water. The rope is 12 m long and is assumed to obey Hooke's law. What should the spring constant of the rope be if the woman is to just reach the water? Homework Equations Ep=mgh E=1/2 kx^2 The...
  26. Blockade

    B Why is there a negative in Hooke's Law (F = -kx)

    For Hooke's Law, why is there a negative in F = -k*x? Is it because k is the reaction force against the force applied on the spring and not the force pushing or pulling on it? For example, let's say that the origin is set where the spring ends when it's in equilibrium with no forces applied to...
  27. K

    Uniaxial Tension Test

    I conducted a uniaxial tension test for a variety of materials but wasn't able to gather much useful axial strain data due to the extensometer continually slipping. I have axial strain data for the linear elastic region of the stress strain curve and I also have the extension of the crossbar of...
  28. C

    Centrifugal force and elastic deformation

    Homework Statement Consider a spring of natural length L_0 with constant k which rests on a horizontal frictionless surface. The spring is attached at one end to a fixed post and at the other end to a mass m. Suppose the spring is rotating around the post in a circle with angular velocity w...
  29. C

    Rod rotating about pivot with spring

    Homework Statement Consider the following classic problem: we have a rod in the vertical position with a pivot at its midpoint and a spring attached to the bottom of the rod, perpendicular to the rod. The is rotated through a small angle theta to the vertical, and released. Find the period of...
  30. S

    Spring Constant in Hooke's Law

    How does one arrive at the following equation to approximate spring constant for solids... using Hooke's Law F ∝-x ⇒ F = -kx and strain∝stress ? k = (m/a2) × (K/ρ)½ where k≡spring constant m ≡ mass of a single atom a ≡ atomic spacing K ≡ bulk modulus ρ ≡ density
  31. M

    Hooke's Law and plastic materials

    Hi, About elastic and plastic materials: All materials exhibit elastic deformation up to a certain limit, beyond which they exhibit plastic deformation. Some materials, such as plasticine, have extremely tiny elastic regions, so we call them 'plastic materials'. Some, like rubber, have large...
  32. Chrono G. Xay

    Calculate the 'Feel' of a Drumhead?

    As another of my personal music projects, I have wondered if it would not be possible to calculate the 'feel' of a drumhead (i.e. the amount of 'give' expressed as transverse displacement 'z' that an equally pre-tensioned circular membrane of radius 'r' experiences when struck on its plane at a...
  33. D

    Simple Hooke's Law Question

    I am new here and I hope I posted this in the right place. I understand that Hooke's Law is F=-ks which states that the spring force is directly proportional but opposite to the difference of the distance from the stretched length of the string and the equilibrium length of the string. I just...
  34. D

    Need to find the source for a 3D Hooke's law equation to cite

    Homework Statement I am using the 3D implementation of Hooke's law as found in wikipedia at: https://en.wikipedia.org/wiki/Hooke%27s_law I need to find the actual source to cite it in a master's project software write up. Many professors do not consider wikipedia to be an acceptable source...
  35. Remixex

    Hooke's and Newton's law to find Second order ODE

    Homework Statement A weight of 8 pounds extends a spring 2 feet. It's assumed that the damping force that acts on the system is equal (number-wise) to alpha times the speed of the weight. Determine the value of alpha > zero so x(t) is critically damped. Determine x(t) if the weight is liberated...
  36. 3

    Hooke's law in tensor form

    Hi! I'm studying physics and currently taking the first mechanics course. After dealing with rotation and gyroscopes, now we're working on things like shear stress, and Hooke's law in tensor form etc. I've got Kleppner/Kolenkow but shear stress, Hooke's law in tensor form and tensors in...
  37. kelvin490

    Same E for true stress true strain?

    Before yield we know that stress/strain= E. Can we obtain the same E if we substitute true stress and true strain in the equation? Why
  38. thegreengineer

    Hooke's law confusion

    Well, I know that Hooke's law establishes that the force applied on a spring is proportional to the displacement. However, I've got a little bit confused about the formula. My textbook manages the formula as the following: F=-kx Whereas some websites manage it as this: F=kx I still don't...