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.
I'm trying to understand the math behind Hooke's law and work done by a spring. I'm really looking for clarity. I am trying to understand why after integrating the equation I get three different results. I understand that each formula is specific to a problem but why does this work...
Homework Statement
An unstrained horizontal spring has a length of 0.32 m and a spring constant of 220 N/m. Two small charged objects are attached to this spring, one at each end. the charges on the objects have equal magnitudes. Because of these charges, the spring stretches 0.020 m, relative...
A force of 16,000lb compresses a string from its natural length of 13 inch to 8 inch. Find the work done to compress it to the first inch
W=$$\int F dx$$
F=kx
16000=K(5)
3200=KW=$$\int F dx$$
W=$$\int_1^{13}\!\ 3200xdx$$
[1600x^2] from 1 to 13
w= 268800ftlb
Am I right?
I think the trick...
Hi,
I know that for a massless spring the tension is T = kx, where k is a constant and x is the compression or elongation.
What is the expression of the tension if the spring has a mass m?
Thanks in advance.
Hello,
I am self studying physics and am a bit confused with Hooke's law.
Using the conservation of energy equation:
(PE+KE)_i = (PE+KE)_f
I don't see why the springs potential energy (½ kx^2) goes in the final part.
Could someone please explain?
EDIT: Also, (I'm a new user) can you...
Homework Statement
An engineer is designing a spring to be placed at the bottom of an elevator shaft. If the elevator cable should happen to break when the elevator is at a height h above the top of the spring, calculate the value of the spring constant k so that the passengers undergo an...
Homework Statement
As shown in Figure A, a block of mass m is hanging from a spring attached to the ceiling. As shown in Figure B, two blocks of mass m/2 are hanging from two strings that are attached to a spring that has the same spring constant k. If the spring in Figure A is stretched a...
Calculus 2- Water pump problem
Homework Statement
The lower half of a sphere with radius 2m is filled with water. Find the work needed to pump the water to a point 1m above the centre of the sphere.
Homework Equations
The Attempt at a Solution
I know you have to cut the problem...
Homework Statement
A 46g rock is placed in a slingshot and the rubber band is stretched. The force of the rubber band on the rock is shown by the graph in the figure(Figure 1) .
http://session.masteringphysics.com/problemAsset/1073689/4/10.P37.jpg
a)Is the rubber band stretched to the...
Homework Statement
F=kx
What does k represent?
Homework Equations
The Attempt at a Solution
k is the proportionality constant of restoring force over displacement. Or with respect to displacement?
Hi all,
I am reading an online material on elastic force and Hooke's law on spring. The definition of the Hooke's law reads that the restoring force is linear proportional to the displacement of the spring with constant k. The restoring force is defined as the force bringing the object back...
Homework Statement
A 50kg mass is attached to a spring and hung from an overhead beam. The Force on the spring when extended 2 meters from rest is 50N. The resting length of the spring is 1m.
1) Obtain the ODE to solve for the velocity as a function of position (NOT time)
2) Solve the...
Hi,
For a linear spring one has Hooke's law:
F = k x
Is there an equivalent law for a spherical elastic balloon giving pressure inside as a function of radius?
Thanks,
John
Hello all,
(Newbie here so I hope I'm posting this in the right section, if not please guide me to the correct one)
I'm studying Hooke's Law in Physics and there is something I can't understand.
The equation for the Simple Harmonic Motion of a spring is given by x'' + wx =0
the general...
Homework Statement
Which one obeys Hooke’s Law and why?
For the material(s) that did obey Hooke’s Law, calculate the spring constant, k. SHOW ALL WORK: listed variables, units, equations without substitutions, equations with substitutions, final answer with units!
The Green line on the...
Hi guys - we did an experiment at college putting some weights on a spring for our Hooke's Law module. When I graph it I get a value for c, in the equation y=mx+c. Is this correct? Will c have a value?
Thanks.
Homework Statement
At a waterpark, sleds with riders are sent along a slippery,
horizontal surface by the release of a large compressed spring. The
spring with force constant and negligible mass
rests on the frictionless horizontal surface. One end is in contact
with a stationary wall. A...
There is a statement in my book i can't really understand. "To stretch a spring, we must do work. We apply equal and opposite forces to the ends of a spring and gradually increase the forces. We hold the left end stationary, so the force we apply at this end does no work.The force at the moving...
I did an experiment where I measured the stretch on a spring from its initial position when masses were added on the end of it, in increments of 50g. I plotted a graph of Force against length stretched and the graph does not go through the origin, or at least it is not linear for forces below...
Homework Statement
A 1.01 kg mass is attached to a spring of force constant 9.5 N/cm and placed on a frictionless surface. By how much will the spring stretch if the mass moves along a circular path of radius 0.485 m at a rate of 2.14 revolutions per second?
What i was thinking was using the...
Homework Statement
A string of natural length 10 in. stretches 1.5 in. under a weight of 8-lb. Find the work done in stretching the spring from its natural length to a length of 14 in.Homework Equations
\begin{equation}
W= \int_a^b \, f(x)dx
\end{equation}
\begin{equation}
f(x)=kx...
Homework Statement
The problem is in the attachment
Homework Equations
F=-kx
k1x1=k2x1
The Attempt at a Solution
I got as my answer Keq= (k1k2)/(k1+k2)
i just wanted to see if i had the right idea or no.
I understand the simplest application of the summation convention.
x_{i}y_{i}
I create a sum of terms such that in each term the subscripts are the same i.e.
x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}+...
But now when I look at understanding summation convention applied to the generalised Hooke's law...
Hooke's law describes a certain light spring of unstretched length 37.0 cm. When one end is attached to the top of a door frame and a 8.00-kg object is hung from the other end, the length of the spring is 40.5 cm.
(a) Find its spring constant.
(b) The load and the spring are taken down. Two...
Do you see where it says
m(dv/dt)v = (d/dt)(.5mv^2)
If it's an integration which I don't think it is then I would think it should be
(d/dt)((mv^3)/3) because you're taking the two v's and adding an additional power. I don't think it is an integration because it says right there in the...
Homework Statement
Hi I know for Hooke's Law F=-kx where -ve sign implies a restoring force back to equilibrium in the opposite direction of x.
My question is:
I have a mass being displaced attached to a spring - consider M2 and z2(t) and K2 in the link below...
I was just thinking, shouldn't hooke's law be represented with an exponential equation?
I am thinking that in class, the springs we use for the hooke's law experiment are small, but if they were very large, then the higher coils would be sustaining much more weight than the lower coils, and...
This was on a practice test for an upcoming exam.
You have persuaded your friend Astrid to attempt an illegal bungee jump from a Bridge. You will provide the bungee cord which has a total length of 40 m and a spring constant of k = 16 N/m. During the jump, Astrid will first fall freely for a...
Homework Statement
Seven and one half foot pounds of work is required to compress a spring 2 inches from its natural length. Find the work required to compress the spring an additional one-half inch
Homework Equations
W=kd
W=∫f(x)dx
The Attempt at a Solution
7.5=W
∫kxdx from...
Homework Statement
Use Hooke's Law to determine the variable force in spring problem.
Eighteen foot-pounds of work is required to stretch a spring 4 inches from it's natural length. Find the work required to stretch the spring an additional 3 inches.
Homework Equations
W=∫f(x)dx
f=kd...
Hello all,
I am currently working at a problem that is (seems) simple at calculations but I am puzzled by the word problems.
The problems are:
1. A spring powered projectile launcher contains a spring with:
k=300 N/m
The spring is compressed by 15.0 cm.
a. How much energy is...
Homework Statement
A mass M = 0.454 kg moves with an initial speed v = 2.88 m/s on a level frictionless air track. The mass is initially a distance D = 0.250 m away from a spring with k = 876 N/m, which is mounted rigidly at one end of the air track. The mass compresses the spring a maximum...
Homework Statement
I am currently doing some Hooke's Law problems. While I do not have any trouble with any exercise in particular, I do have trouble with the sign in the equation. Let's say I have a vertical spring and I attached a hanging mass m to it. The string will then stretch a distance...
Homework Statement
4 springs with a mass on each end are connected in series as below:
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m1
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m2
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m3
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m4
All the masses are mass m, the length of each spring is 1, and the spring constant is k, find the extension of each spring...
Depending on how you fall, you can break a bone easily. The severity of the break depends on how much energy the bone absorbs in the accident, and to evaluate this let us treat the bone as an ideal spring. The maximum applied force of compression that one man’s thighbone can endure without...
Homework Statement
A vertical ideal spring is mounted on the floor and has a spring constant of 132 N/m. A 0.80-kg block is placed on the spring in two different ways. (a) In one case, the block is placed on the spring and not released until it rests stationary on the spring in its equilibrium...
Homework Statement
A 2.70kg mass is pushed against a horizontal spring of force constant 25.0 N/cm on a frictionless air table. The spring is attached to the tabletop, and the mass is not attached to the spring. When the spring has been compressed enough to store 12.0 J of potential energy...
Homework Statement
I'm given the mass of the object and acceleration (gravity), and I'm given the pull back distance of one test of the slingshot. I've found the k constant by finding force with f=ma, but I need to find the pull back distance using the same slingshot in order to propel the...
Homework Statement
Assume the spring force of a bowstring acting on an arrow does not follow Hooke’s Law because the bow’s material becomes more rigid as it is drawn back. The string’s force as a function of the drawback distance in meters is:
F(x) = -k1x -k2x|x|
where positive x is along...
Homework Statement
A student measures the force required to stretch a spring by various amounts and makes the graph shown in the figure, which plots this force as a function of the distance the spring has stretched.
Part A) Does this spring obey Hooke's Law?
Part B) What is the force...
A lift is used to lower miners 4000m into a mine. The lift is supported by many high strength steel strands of 3mm diameter each. The ultimate stress for the steel strands is 1200MPa and the allowable stress is 600MPa. The empty lift weighs 15kN. It has a capacity of 20 persons. The density of...
Homework Statement
Whether or not the spring is vertical or not, assuming that the spring is ideal, is Fnet always equal to -kx?
Is it incorrect to write the sum of the forces of a vertical spring as Fnet = -kx - mg?
And does hooke's law account for the force of friction on a horizontal...
Homework Statement
A 50g mass hangs at the end of a hookean spring. When a 20g more is added to the spring, it stretches 7.0cm more. Find the spring constant.
The Attempt at a Solution
Isn't it just
mg = kx
mg/x = k
Shouldn't m = 70g instead of 20g?
My key says m = 20g?
Homework Statement
A 0.75 kg block is held at rest against a spring compressing it 27.0 cm. The spring constant is 735 N/m. The block is released and it begins to move toward a loop across a frictionless surface. How much kinetic energy does the block have when the spring is compressed 13.5...
Homework Statement
is the hooke's constant same for compression as well as extension of all the materials and springs?
Homework Equations
Force = hooke's constant * extension [ to a limit]
The Attempt at a Solution
Homework Statement
A ball with mass 'm' is pushed against a spring (with a spring constant 'k') and causes a displacement 'x' on the spring. Once the force is released from, the ball is shot horizontally. What is the velocity 'v' horizontally?
Homework Equations
E(k) = 1/2 mv^2
E(s) =...
Homework Statement
A small truck is equipped with a rear bumper that has a spring constant of 800,000 N/m. the bumper can be compressed up to 15cm without causing damage to the truck. What is the maximum velocity with which a solid 1000-kg car can collide with the bumper without causing damage...
A 2.0 kg mass and a 3.0 kg mass are on a horizontal frictionless surface, connected by a massless spring with spring constant k=140N/m. A 15 N force is applied to the larger mass, as a shown (see picture). How much does the spring stretch from its equilibrium length?
I've solved...