Young s modulus. W.E. Bailey, APAM/MSE EN1102

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1 Young s modulus W.E. Bailey, APAM/MSE EN1102

2 Spring constants Remember k is the spring constant Consider two springs F = k x Figure: Thin wire (d 1 cm) W.E. Bailey, APAM/MSE EN1102 Young s modulus 2 / 8

3 Spring constants Figure: Thick wire (d 1 m) W.E. Bailey, APAM/MSE EN1102 Young s modulus 3/8

4 Spring constants are different Wire between towers L 2.5 mm displacement over L =100 m for 50 kg W.E. Bailey, APAM/MSE EN1102 Young s modulus 4 / 8

5 Spring constants are different Wire between towers L 2.5 mm displacement over L =100 m for 50 kg Wires to New Jersey L 400 mm displacement over L =1000 m for kg (weight of wire) So, estimating: W.E. Bailey, APAM/MSE EN1102 Young s modulus 4 / 8

6 Spring constants are different Wire between towers L 2.5 mm displacement over L =100 m for 50 kg Wires to New Jersey L 400 mm displacement over L =1000 m for kg (weight of wire) So, estimating: Case 1: k Petit = 500 N/ 2.5 mm: k Petit = N/m W.E. Bailey, APAM/MSE EN1102 Young s modulus 4 / 8

7 Spring constants are different Wire between towers L 2.5 mm displacement over L =100 m for 50 kg Wires to New Jersey L 400 mm displacement over L =1000 m for kg (weight of wire) So, estimating: Case 1: k Petit = 500 N/ 2.5 mm: k Petit = N/m Case 2: k GWB = N/ 0.4 m = k GWB = N/m W.E. Bailey, APAM/MSE EN1102 Young s modulus 4 / 8

8 Spring constants are different Wire between towers L 2.5 mm displacement over L =100 m for 50 kg Wires to New Jersey L 400 mm displacement over L =1000 m for kg (weight of wire) So, estimating: Case 1: k Petit = 500 N/ 2.5 mm: k Petit = N/m Case 2: k GWB = N/ 0.4 m = k GWB = N/m Both wires are steel! W.E. Bailey, APAM/MSE EN1102 Young s modulus 4 / 8

9 Spring constants are different Wire between towers L 2.5 mm displacement over L =100 m for 50 kg Wires to New Jersey L 400 mm displacement over L =1000 m for kg (weight of wire) So, estimating: Case 1: k Petit = 500 N/ 2.5 mm: k Petit = N/m Case 2: k GWB = N/ 0.4 m = k GWB = N/m Both wires are steel! Spring constant not very predictive... W.E. Bailey, APAM/MSE EN1102 Young s modulus 4 / 8

10 Define different quantities Stress σ F A F : wire tension, N A: wire area, m 2 Dimensions of pressure: N/m 2 [=] Pa W.E. Bailey, APAM/MSE EN1102 Young s modulus 5 / 8

11 Define different quantities Stress σ F A F : wire tension, N A: wire area, m 2 Dimensions of pressure: N/m 2 [=] Pa Strain ɛ L L W.E. Bailey, APAM/MSE EN1102 Young s modulus 5 / 8

12 Define different quantities Stress σ F A F : wire tension, N A: wire area, m 2 Dimensions of pressure: N/m 2 [=] Pa Strain ɛ L L Percentage change in length (dimensionless) W.E. Bailey, APAM/MSE EN1102 Young s modulus 5 / 8

13 Elastic modulus Related through Young s modulus (elastic) E σ = Eɛ W.E. Bailey, APAM/MSE EN1102 Young s modulus 6 / 8

14 Elastic modulus Related through Young s modulus (elastic) E σ = Eɛ E has units of GPa Values: Rubber (styrene-butadiene): GPa W.E. Bailey, APAM/MSE EN1102 Young s modulus 6 / 8

15 Elastic modulus Related through Young s modulus (elastic) E σ = Eɛ E has units of GPa Values: Rubber (styrene-butadiene): GPa Gold (Au): 77 GPa W.E. Bailey, APAM/MSE EN1102 Young s modulus 6 / 8

16 Elastic modulus Related through Young s modulus (elastic) E σ = Eɛ E has units of GPa Values: Rubber (styrene-butadiene): GPa Gold (Au): 77 GPa Steel (Fe-C), 207 GPa W.E. Bailey, APAM/MSE EN1102 Young s modulus 6 / 8

17 Elastic modulus Related through Young s modulus (elastic) E σ = Eɛ E has units of GPa Values: Rubber (styrene-butadiene): GPa Gold (Au): 77 GPa Steel (Fe-C), 207 GPa Tungsten (W): 400 GPa W.E. Bailey, APAM/MSE EN1102 Young s modulus 6 / 8

18 Spring constant from Young s modulus Recall the spring constant, F = k x W.E. Bailey, APAM/MSE EN1102 Young s modulus 7 / 8

19 Spring constant from Young s modulus Recall the spring constant, F = k x Substituting in with σ = F /A and ɛ = l/l 0 = x/l 0, σa = kɛl 0 W.E. Bailey, APAM/MSE EN1102 Young s modulus 7 / 8

20 Spring constant from Young s modulus Recall the spring constant, F = k x Substituting in with σ = F /A and ɛ = l/l 0 = x/l 0, σa = kɛl 0 From the definition of the Young s modulus, σ = Eɛ, EA = kl 0 W.E. Bailey, APAM/MSE EN1102 Young s modulus 7 / 8

21 Spring constant from Young s modulus Recall the spring constant, F = k x Substituting in with σ = F /A and ɛ = l/l 0 = x/l 0, σa = kɛl 0 From the definition of the Young s modulus, σ = Eɛ, EA = kl 0 E = k l 0 A W.E. Bailey, APAM/MSE EN1102 Young s modulus 7 / 8

22 Spring constant from Young s modulus Recall the spring constant, F = k x Substituting in with σ = F /A and ɛ = l/l 0 = x/l 0, σa = kɛl 0 From the definition of the Young s modulus, σ = Eɛ, EA = kl 0 E = k l 0 A W.E. Bailey, APAM/MSE EN1102 Young s modulus 7 / 8

23 Spring constants are different, Young s modulus the same Wire between towers L 2.5 mm displacement, A =1 cm 2, L 0 =100 m W.E. Bailey, APAM/MSE EN1102 Young s modulus 8 / 8

24 Spring constants are different, Young s modulus the same Wire between towers L 2.5 mm displacement, A =1 cm 2, L 0 =100 m k Petit = N/m W.E. Bailey, APAM/MSE EN1102 Young s modulus 8 / 8

25 Spring constants are different, Young s modulus the same Wire between towers L 2.5 mm displacement, A =1 cm 2, L 0 =100 m k Petit = N/m l 0 E = k Petit A = N/m 100 m 2 = 200 GPa 1 cm W.E. Bailey, APAM/MSE EN1102 Young s modulus 8 / 8

26 Spring constants are different, Young s modulus the same Wire between towers L 2.5 mm displacement, A =1 cm 2, L 0 =100 m k Petit = N/m l 0 E = k Petit A = N/m 100 m 2 = 200 GPa 1 cm Wires to New Jersey L 400 mm displacement, A =1 m 2, L 0 =1000 m W.E. Bailey, APAM/MSE EN1102 Young s modulus 8 / 8

27 Spring constants are different, Young s modulus the same Wire between towers L 2.5 mm displacement, A =1 cm 2, L 0 =100 m k Petit = N/m l 0 E = k Petit A = N/m 100 m 2 = 200 GPa 1 cm Wires to New Jersey L 400 mm displacement, A =1 m 2, L 0 =1000 m k GWB = N/m W.E. Bailey, APAM/MSE EN1102 Young s modulus 8 / 8

28 Spring constants are different, Young s modulus the same Wire between towers L 2.5 mm displacement, A =1 cm 2, L 0 =100 m k Petit = N/m l 0 E = k Petit A = N/m 100 m 2 = 200 GPa 1 cm Wires to New Jersey L 400 mm displacement, A =1 m 2, L 0 =1000 m k GWB = N/m l 0 E = k GWB A = N/m 1 km 2 = 200 GPa 1 m W.E. Bailey, APAM/MSE EN1102 Young s modulus 8 / 8

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