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Learning objectives

By the end of this section, you will be able to:

  • State Hooke's law.
  • Explain Hooke's law using graphical representation between deformation and applied force.
  • Discuss the three types of deformations such as changes in length, sideways shear, and changes in volume.
  • Describe with examples the Young's modulus, shear modulus, and bulk modulus.
  • Determine the change in length given mass, length, and radius.

We now move from consideration of forces that affect the motion of an object (such as friction and drag) to those that affect an object's shape. If a bulldozer pushes a car into a wall, the car will not move but it will noticeably change shape. A change in shape due to the application of a force is a deformation    . Even very small forces are known to cause some deformation. For small deformations, two important characteristics are observed. First, the object returns to its original shape when the force is removed—that is, the deformation is elastic for small deformations. Second, the size of the deformation is proportional to the force—that is, for small deformations, Hooke's law is obeyed. In equation form, Hooke's law    is given by

F = k Δ L , size 12{F=kΔL} {}

where Δ L size 12{ΔL} {} is the amount of deformation (the change in length, for example) produced by the force F size 12{F} {} , and k size 12{k} {} is a proportionality constant that depends on the shape and composition of the object and the direction of the force. Note that this force is a function of the deformation Δ L size 12{ΔL} {} —it is not constant as a kinetic friction force is. Rearranging this to

Δ L = F k size 12{ΔL= { {F} over {k} } } {}

makes it clear that the deformation is proportional to the applied force. [link] shows the Hooke's law relationship between the extension Δ L size 12{ΔL} {} of a spring or of a human bone. For metals or springs, the straight line region in which Hooke's law pertains is much larger. Bones are brittle and the elastic region is small and the fracture abrupt. Eventually a large enough stress to the material will cause it to break or fracture.

Hooke's law

F = kΔL , size 12{F=kΔL} {}

where Δ L size 12{ΔL} {} is the amount of deformation (the change in length, for example) produced by the force F size 12{F} {} , and k size 12{k} {} is a proportionality constant that depends on the shape and composition of the object and the direction of the force.

Δ L = F k size 12{ΔL= { {F} over {k} } } {}
Line graph of change in length versus applied force. The line has a constant positive slope from the origin in the region where Hooke's law is obeyed. The slope then decreases, with a lower, still positive slope until the end of the elastic region. The slope then increases dramatically in the region of permanent deformation until fracturing occurs.
A graph of deformation Δ L size 12{ΔL} {} versus applied force F size 12{F} {} . The straight segment is the linear region where Hooke's law is obeyed. The slope of the straight region is 1 k size 12{ { {1} over {k} } } {} . For larger forces, the graph is curved but the deformation is still elastic— Δ L size 12{ΔL} {} will return to zero if the force is removed. Still greater forces permanently deform the object until it finally fractures. The shape of the curve near fracture depends on several factors, including how the force F size 12{F} {} is applied. Note that in this graph the slope increases just before fracture, indicating that a small increase in F size 12{F} {} is producing a large increase in L size 12{L} {} near the fracture.

The proportionality constant k size 12{k} {} depends upon a number of factors for the material. For example, a guitar string made of nylon stretches when it is tightened, and the elongation Δ L size 12{ΔL} {} is proportional to the force applied (at least for small deformations). Thicker nylon strings and ones made of steel stretch less for the same applied force, implying they have a larger k size 12{k} {} (see [link] ). Finally, all three strings return to their normal lengths when the force is removed, provided the deformation is small. Most materials will behave in this manner if the deformation is less than about 0.1% or about 1 part in 10 3 size 12{"10" rSup { size 8{3} } } {} .

Questions & Answers

Propose a force standard different from the example of a stretched spring discussed in the text. Your standard must be capable of producing the same force repeatedly.
Giovani Reply
What is meant by dielectric charge?
It's Reply
what happens to the size of charge if the dielectric is changed?
Brhanu Reply
omega= omega not +alpha t derivation
Provakar Reply
u have to derivate it respected to time ...and as w is the angular velocity uu will relace it with "thita × time""
Abrar
do to be peaceful with any body
Brhanu Reply
the angle subtended at the center of sphere of radius r in steradian is equal to 4 pi how?
Saeed Reply
if for diatonic gas Cv =5R/2 then gamma is equal to 7/5 how?
Saeed
define variable velocity
Ali Reply
displacement in easy way.
Mubashir Reply
binding energy per nucleon
Poonam Reply
why God created humanity
Manuel Reply
Because HE needs someone to dominate the earth (Gen. 1:26)
Olorunfemi
why god made humenity
Ali
Is the object in a conductor or an insulator? Justify your answer. whats the answer to this question? pls need help figure is given above
Jun Reply
ok we can say body is electrically neutral ...conductor this quality is given to most metalls who have free electron in orbital d ...but human doesn't have ...so we re made from insulator or dielectric material ... furthermore, the menirals in our body like k, Fe , cu , zn
Abrar
when we face electric shock these elements work as a conductor that's why we got this shock
Abrar
how do i calculate the pressure on the base of a deposit if the deposit is moving with a linear aceleration
ximena Reply
why electromagnetic induction is not used in room heater ?
Gopi Reply
room?
Abrar
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Amoah Reply
What is law of gravition
sushil Reply
Practice Key Terms 6

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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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