PHY2053 Lecture 21 Ch. 10.1-10.4: Elastic Deformation, Shear

PHY2053, Lecture 4, Motion in a Plane

Definition of Strain• need good variables to represent the cause

impacting upon the material (example: wire) and the response of the material

• system response:• consider a wire of length L, when pulled

by force F, it will stretch by ∆L• if we subject a wire of length 2L to the

same force, it is reasonable to expect it to stretch by 2 ∆L

• define strain as measure of system response:

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PHY2053, Lecture 4, Motion in a Plane

Definition of Stress• similar situation, now imagine using a wire

with twice the diameter• intuitively expect the extension to be less,

despite the same force• in reality, force is distributed across cross-

sectional area, not the diameter of the wire• not surprisingly, one finds that the system

response is proportional to F/A• define stress as cause acting upon system

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PHY2053, Lecture 4, Motion in a Plane

Full definition of Hooke’s Law• previously we have used the formula:

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to describe elastic deformations (F was the force with which the system is resisting change, ∆x its extension)

• a more universal, material-specific definition is the full definition of Hooke’s Law:

• in this convention, F is the force which is acting on the system (hence the change of sign between formulas)

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PHY2053, Lecture 4, Motion in a Plane

Example #1:• A wire of length 5 m and cross sectional area of 0.1

cm2 stretches by 6.5 mm when a load of 1 kN is hung from it. What is the Young’s modulus for this wire?

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PHY2053, Lecture 4, Motion in a Plane

H-ITT: Steel WireA steel wire with Young’s modulus 21×1010 Pa is 1 m long and has a cross-sectional area of 1 mm2. In order to be used in a piano to produce the middle C, it has to have a tension of 5.40 kN. How much does the tuner have to stretch the wire in order to tune it to middle C?

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A) roughly 5.2 cmB) roughly 1.3 cmC) is quite definitely the wrong answerD) roughly 0.65 cmE) roughly 2.6 cm

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PHY2053, Lecture 4, Motion in a Plane

H-ITT: Steel Wire, Discussion

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PHY2053, Lecture 4, Motion in a Plane

H-ITT: Brass WireA brass wire with Young’s modulus 9×1010 Pa and density ρ=8400 kg/m3 is strung in a piano frame so that its unstrained length is 1.5 m. In order to produce a tone of frequency f, the tension in the string has to be:

area of the string. By how much does the string have to be stretched to produce the C2 tone (65.4 Hz)?

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A) roughly 13.0 mmB) roughly 4.2 mmC) roughly 1.4 mmD) roughly 65 μmE) is most certainly the wrong answer

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PHY2053, Lecture 4, Motion in a Plane

H-ITT: Brass Wire, Discussion

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10PHY2053, Lecture 4, Motion in a Plane

Hooke’s Law for Shear Deformations• shear deformation is the result of a pair of equal and

opposite forces that act parallel to opposite surfaces• define action upon system - shear stress:

• and the system response - shear strain:

• Hooke’s law for shear deformations connects the two:

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PHY2053, Lecture 4, Motion in a Plane

Example #2:• The upper surface of a sponge, 5 cm on each side, is

displaced by 2 cm by a tangential force. If the shear modulus of the sponge is 2000 Pa, what is the magnitude of the tangential force?

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PHY2053, Lecture 4, Motion in a Plane 12

Hooke’s Law for Volume Deformations• consider an object immersed in a fluid - the fluid will

be exerting pressure from all sides• this can also cause deformations of the system• action upon the system in this case → volume stress

• and the system response - volume strain:

• Hooke’s law for volume deformations:

Usually ∆P wrt atmospheric

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PHY2053, Lecture 4, Motion in a Plane

Example #3:• By what percentage does the density of water

increase at a depth of 1.0 km below the surface? The bulk modulus for water is 2.2×109 Pa.

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Next LectureSimple Harmonic Oscillations

PHY2053, Lecture 4, Motion in a Plane

Demos

• shear strain• Hooke’s law with wire extension

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