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Diagram of weight w attached to each of three guitar strings of initial length L zero hanging vertically from a ceiling. The weight pulls down on the strings with force w. The ceiling pulls up on the strings with force w. The first string of thin nylon has a deformation of delta L due to the force of the weight pulling down. The middle string of thicker nylon has a smaller deformation. The third string of thin steel has the smallest deformation.
The same force, in this case a weight ( w size 12{w} {} ), applied to three different guitar strings of identical length produces the three different deformations shown as shaded segments. The string on the left is thin nylon, the one in the middle is thicker nylon, and the one on the right is steel.

Stretch yourself a little

How would you go about measuring the proportionality constant k size 12{k} {} of a rubber band? If a rubber band stretched 3 cm when a 100-g mass was attached to it, then how much would it stretch if two similar rubber bands were attached to the same mass—even if put together in parallel or alternatively if tied together in series?

We now consider three specific types of deformations: changes in length (tension and compression), sideways shear (stress), and changes in volume. All deformations are assumed to be small unless otherwise stated.

Changes in length—tension and compression: elastic modulus

A change in length Δ L size 12{ΔL} {} is produced when a force is applied to a wire or rod parallel to its length L 0 size 12{L rSub { size 8{0} } } {} , either stretching it (a tension) or compressing it. (See [link] .)

Figure a is a cylindrical rod standing on its end with a height of L sub zero. Two vectors labeled F extend away from each end. A dotted outline indicates that the rod is stretched by a length of delta L. Figure b is a similar rod of identical height L sub zero, but two vectors labeled F exert a force toward the ends of the rod. A dotted line indicates that the rod is compressed by a length of delta L.
(a) Tension. The rod is stretched a length Δ L size 12{ΔL} {} when a force is applied parallel to its length. (b) Compression. The same rod is compressed by forces with the same magnitude in the opposite direction. For very small deformations and uniform materials, Δ L size 12{ΔL} {} is approximately the same for the same magnitude of tension or compression. For larger deformations, the cross-sectional area changes as the rod is compressed or stretched.

Experiments have shown that the change in length ( Δ L size 12{ΔL} {} ) depends on only a few variables. As already noted, Δ L size 12{ΔL} {} is proportional to the force F size 12{F} {} and depends on the substance from which the object is made. Additionally, the change in length is proportional to the original length L 0 size 12{L rSub { size 8{0} } } {} and inversely proportional to the cross-sectional area of the wire or rod. For example, a long guitar string will stretch more than a short one, and a thick string will stretch less than a thin one. We can combine all these factors into one equation for Δ L size 12{ΔL} {} :

Δ L = 1 Y F A L 0 , size 12{ΔL= { {1} over {Y} } { {F} over {A} } L rSub { size 8{0} } } {}

where Δ L size 12{ΔL} {} is the change in length, F size 12{F} {} the applied force, Y size 12{Y} {} is a factor, called the elastic modulus or Young’s modulus, that depends on the substance, A size 12{A} {} is the cross-sectional area, and L 0 size 12{L rSub { size 8{0} } } {} is the original length. [link] lists values of Y size 12{A} {} for several materials—those with a large Y size 12{A} {} are said to have a large tensile stifness because they deform less for a given tension or compression.

Elastic moduli Approximate and average values. Young’s moduli Y size 12{Y} {} for tension and compression sometimes differ but are averaged here. Bone has significantly different Young’s moduli for tension and compression.
Material Young’s modulus (tension–compression) Y ( 10 9 N/m 2 ) Shear modulus S ( 10 9 N/m 2 ) Bulk modulus B ( 10 9 N/m 2 )
Aluminum 70 25 75
Bone – tension 16 80 8
Bone – compression 9
Brass 90 35 75
Brick 15
Concrete 20
Glass 70 20 30
Granite 45 20 45
Hair (human) 10
Hardwood 15 10
Iron, cast 100 40 90
Lead 16 5 50
Marble 60 20 70
Nylon 5
Polystyrene 3
Silk 6
Spider thread 3
Steel 210 80 130
Tendon 1
Acetone 0.7
Ethanol 0.9
Glycerin 4.5
Mercury 25
Water 2.2

Young’s moduli are not listed for liquids and gases in [link] because they cannot be stretched or compressed in only one direction. Note that there is an assumption that the object does not accelerate, so that there are actually two applied forces of magnitude F size 12{F} {} acting in opposite directions. For example, the strings in [link] are being pulled down by a force of magnitude w size 12{w} {} and held up by the ceiling, which also exerts a force of magnitude w size 12{w} {} .

Questions & Answers

Calculate the work done by an 85.0-kg man who pushes a crate 4.00 m up along a ramp that makes an angle of 20.0º20.0º with the horizontal. (See [link] .) He exerts a force of 500 N on the crate parallel to the ramp and moves at a constant speed. Be certain to include the work he does on the crate an
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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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