What is Hooke's law: Definition and 260 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. JMAMA

    Hooke's Law using Potential Energy

    Max speed occurs when all energy has been translated from spring into box. E (Potential) = 1/2kx^2 E (Potential) = (1/2)(42 N/m)(0.280 m)^2 = 1.6464 N m Ep = Ek =1/2mv^2 1.6464 N m= 1/2 (1.2 kg) v^2 v = 1.6565 m/s
  2. O

    2 masses connected by spring, one is pulled, how much does the spring stretch?

    Hi, I am having trouble with this problem. I'm thinking the solution is this but I'm not sure. Fnet=m1a+m2aFnet=m1a+m2a , m1a=kxm1a=kx, m2a=F−kxm2a=F−kx so x=m1ak=−(m2a−F)kx=m1ak=−(m2a−F)k
  3. P

    Problem with when to use Force and Energy. What compresses spring/

    This is how I tried to do it, which is the most direct. The force that the mass exerts on the spring is mgsin(53). and I equated that to kx. and found x. but apparently, this is wrong and the teacher told me a different method. (ME)1=(ME)2 due to conservation of mechanical energy...
  4. HighFlyer

    B Springboard physics questions for diving coach at YMCA

    I'm a diving coach at the local YMCA and I want to give a lesson regarding the physics of diving off the board with maximum efficiency. This is the type of diving board with an adjustable fulcrum, basically a lever with one end fixed with bolts. Its been almost 15 years since I've taken a...
  5. P

    I Thoughts about coupled harmonic oscillator system

    Same instruction was given while finding value of 'g' by a bar pendulum. In the former case,does the spring obeys hooke's law while it forms a coupled harmonic oscillator system?Does the bar pendulum somehow breaks the simple harmonic motion(such that we can't apply the law for sumple harmonic...
  6. nomadreid

    Investigating Unexpected Results in Elementary Experiment

    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...
  7. 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...
  8. Y

    Will the spring stretch or compress?

    i would like to understand why my answer is incorrect i don't know what I am doing wrong i would like for some help
  9. K

    A Hooke's law, Bertrand's theorem and closed orbits

    Bertrand's Theorem says : the only forces whose bounded orbits imply closed orbits are the Hooke's law and the attractive inverse square force. I'm looking at the hookes law ##f=-k r## and try to see explicitly that the orbit is indeed closed. I use the orbit equation ##\frac{d^{2} u}{d...
  10. 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...
  11. 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. ##...
  12. P

    Hooke's law does not apply when there is friction, right?

    Summary:: I am not sure about how can I write an appropriate equation to a question which include friction force and initial velocity. I came across a question that I am not sure how to write an equation.In the question, there is an object of mass m that is thrown into spring at v initial...
  13. patric44

    Question about the nonlinear Hooke's law

    hi guys i saw this problem online about using the MATLAB ode45 to solve the nonlinear Hoock's law and its specifically stated that the nonlinear hoock's law is given by $$F = k\;u + \epsilon\;u^{3}$$ , but when expanding the potential function in a Taylor series where you obtain the force...
  14. jaumzaum

    Real Hooke's law for springs

    I was wondering which equation do springs obey better: $$F=-kx$$ $$F=-k ln(x/x_0)$$ The first is Hooke's law, but the second comes when we consider the relative deformation instead of the absolute deformation. I am asking because I haven't seen any website stating the second equation, I just...
  15. X

    Springs, Hooke's Law, Work, and Energy

    The formulas we have been given include Potential energy=mgh, Stored strain energy=(1/2)K(change in X)^2, , Kinetic energy=(1/2)mV^2, Work=F(change in d), Force=K(change in X). Not sure how exactly to answer the question.
  16. archaic

    B Trying to understand the basics of Hooke's law

    Hello, I need some help on understanding what this book is trying to convey. How does "any part of the spring acts on another part"? Doesn't (2.25) just give us the "operator force" and, since the spring is at equilibrium, the elastic force? What exactly is strain? And I am failing to see the...
  17. J

    Manipulating equation for Young's Modulus and Hooke's Law

    Equating the two equations gives me ##k = -\frac {YA}{L}## but the correct answer of the same magnitude but opposite sign. I think the nub of my misunderstanding is quite elementary (who would have guessed!) : When is it ##F=kx## and when is it ##F=-kx##? If I understand correctly, F is the...
  18. K

    Hooke's Law vs. Conservation of Energy

    Here are the two questions I want to compare: 1. A student of mass 62 kg stands on an upholstered chair containing springs, each of force constant 2.4 × 103 N/m. If the student is supported equally by six springs, what is the compression of each spring? 2. A 0.20-kg ball attached to a vertical...
  19. A

    Work done by Hooke's Law? (Calc 2)

    An open tank has the shape of a right circular cone. The tank is 8 feet across the top and 6 feet high. How much work is done in emptying the tank by pumping the water over the top edge? (The weight-density of water is 62.4 pounds per cubic foot.)
  20. 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/
  21. 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...
  22. 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...
  23. Dr. Courtney

    Insights An Accurate Hooke's Law Laboratory - Comments

    Greg Bernhardt submitted a new blog post An Accurate Hooke's Law Laboratory Continue reading the Original Blog Post.
  24. 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...
  25. F

    B Hooke's Law and Restoring/Applied Force

    According to Hooke's Law, F=-kx where F is the restoring force, k is the spring constant and x is the length of extension/compression. When an applied force compresses a spring, a restoring force will act in the opposite direction. When a spring is compressed and is in equilibrium (not...
  26. 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 = -...
  27. S

    Solving Hooke's Law Problems: Compression & Forces

    Homework Statement A uniform beam AOB, O being the mid point of AB, mass M, rests on three identical vertical springs with stiffness constants k1, k2 and k3 at A, O and B respectively. The bases of the springs are fixed to a horizontal platform. Determine the compression of the springs and...
  28. T

    Help With Understanding Hooke's Law

    Hi all I was hoping someone could help me understand Hookes Law. Hooke’s Law, states that Stress is proportional to Strain. The bit I am struggling to understand is that Stress is F/A and Strain is Change in Length/Original Length, so if Stress is proportional to Strain then shouldn't F/A be...
  29. 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...
  30. Lukeblackhill

    I Hooke's Law: Changes for Large Displacements

    Morning, I've come across this statement in Berkeley Physics Course, Vol.1 - Cp. 5 (pg.149): "For sufficiently small displacements such a force may be produced by a stretched or compressed spring. For large elastic displacements we must add terms in higher powers of x to Eq. (,5.19): Fx = -...
  31. 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...
  32. 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...
  33. mastermechanic

    Video Sources for Anisotropic Hooke's Law

    Hello everyone, I'm taking Mechanics of Materials II this semester which includes Anisotropic Hooke's Law, Plane Stress & Strains, Mohr Circle and so on. I need a video source of these topics. The videos on youtube mostly have bad camera position. I want something like coursera stuff but there...
  34. C

    No prefix: Understanding the Total Force on a 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...
  35. A

    Hooke's Law Lab Spring Constant Calculation

    Homework Statement I need to calculate a spring constant using measurements from a Hooke's Law Apparatus, a spring, and some weights. The weights are hung vertically from the spring and the distance is measured from the equilibrium point of the spring. If I'm solving for k, then k=F/x. I do...
  36. 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...
  37. A

    What is the difference between the yield point and the 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...
  38. 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...
  39. 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 ) ...
  40. S

    Hooke's Law acceleration question

    Homework Statement A small spring with a force constant of 72 N/m is held vertically and then stretched 16 cm. A 1.6kg mass is attached to it and then released. Calculate the acceleration of the mass at the moment of release. Homework Equations F = ma, Ee = 1/2kx^2 The Attempt at a Solution...
  41. C

    Corporate CEO, Physicist

    I am applying a Green's probabilistic elastodynamic tensor with relativistic manifold extensions to solve a pull out of a smoothly shaped deformable spheroid from a stiff inhomogenous deformable quasi-brittle host. This involves a Hooke's law tensor, a relativistic manifold Ricci tensor, a...
  42. Kaneki123

    B A Question about Hooke's Law....

    Okay...Hook's Law is stated as ''the force (F) needed to extend or compress a spring by some distance X is proportional to that distance. That is: F = kX (Wikipedia)'' And further on this topic there is a statement that ''Hooke's law for a spring is often stated under the convention that F is...
  43. J

    Calculating the Spring Constant from Force-Extension Graph

    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...
  44. harini07

    Hooke's law and wave velocity related problem

    Homework Statement The extension in a string, obeying hooke’s law is Y when wave velocity in it is V. if extension is increased to 1.5Y, then wave velocity V’ becomes? 1) V' =V. 2)V'= 1.22V . 3)V'=1.5V. 4) V'=0.75V. Homework Equations wave velocity= frequency*wave length. The Attempt at a...
  45. R

    Hooke's Law Q3: Does Increasing Cross Sectional Area Reduce Work?

    3. The Attempt at a Solution My thinking is that if cross sectional area of the cords increase wouldn't the cords be heavier and thus it would require more work to pull/stretch the device? So more work is done? (But the answers say less work is done because there is a smaller extension/won't...
  46. R

    Hooke's law and elastic deformation

    The Attempt at a Solution [/B] I know that spring A shows elastic deformation since it returned to its original shape once the stress removed and spring c shows plastic deformation and has been permanently extended so it won't return to its original shape. BUT, I'm struggling to complete the...
  47. Kernul

    Hooke's Law Second Order Differential Equation

    Homework Statement A mass ##m## on a frictionless table is connected to a spring with spring constant ##k## so that the force on it is ##F_x = -kx## where ##x## is the distance of the mass from its equilibrium position. It is then pulled so that the spring is stretched by a distance ##x## from...
  48. 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...
  49. L

    Two Springs & Hooke's Law: Effect on Elastic Electrical Lead

    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...
  50. M

    Trying to Calculate k, using Hooke's Law

    Homework Statement The position of a 49 g oscillating mass is given by x(t)=(1.8cm)cos12t, where t is in seconds. Homework Equations k=mg/x The Attempt at a Solution I've tried working this problem multiple different ways and it is just not working for me. I used k= (.049*9.8)/.018 Is this...