What is Young's modulus for this alloy of titanium?

In summary, when a 109 kg mass is hung from a titanium wire with an alloy of titanium, the wire stretches by 1.41 cm. The alloy has a Young's modulus of 4.54 GPa.
  • #1

Homework Statement

A hanging wire made of an alloy of titanium with diameter 0.15 cm is initially 2.8 m long. When a 109 kg mass is hung from it, the wire stretches an amount 1.41 cm. A mole of titanium has a mass of 48 grams, and its density is 4.54 g/cm^3.

Based on these experimental measurements, what is Young's modulus for this alloy of titanium?
From the mass of one mole and the density you can find the length of the interatomic bond (diameter of one atom). This is 2.60E-10 m for titanium. What is the k{s,i}?

Homework Equations

k{s,i}: stiffness of an interatomic bond in a solid.
Y= stress/strain = (tension force/cross section area)/(change of Length/Length)
Y = ((k{s,i}*s)/diameter^2)/(s/diameter) = k{s,i}/diameter

The Attempt at a Solution

stress = changeL/L = .0141m/2.8m = 5.035714E-3
strain = F{T}/A = 9.81*mass / pi*(7.5E-4)^2 = 5.532E11
4.5g/cm^3 = 4.5E3kg/m^3
(mass = Density*length^3 = 4.5E3kg/m^3*(2.8 m)^3 = 99662.08)

stress/strain = Y = 9.10195E-15

I'm pretty sure that's the wrong answer.
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  • #2
Hi badwallpaper0,

The stress is F/A, and the strain is (change in L/L); you seem to have these reversed.

Also, what number did you use for the mass in calculating F/A? I don't see how you got the result 5.532e11.
  • #3
alphysicist said:
Hi badwallpaper0,

The stress is F/A, and the strain is (change in L/L); you seem to have these reversed.

Also, what number did you use for the mass in calculating F/A? I don't see how you got the result 5.532e11.

Hey alphysicist,
First of all, thank you for the help.
Ok, I redid the problem making
mass = density*length^3 = 4.54E3 kg/m^3 * 2.8^3 = 9.966208E4
stress=F{t}/A = 9.81*mass / pi*(7.5E-4)^2 = 5.5325E11
strain=.0141m/2.8m = 5.0357E-3

stress/strain = 5.53256E11/5.0357E-3 = 1.098665E14

I changed the stress and strain as you suggested, and I redid the calculation for mass.
Does that seem like a reasonable/correct answer?
  • #4
I think there are a couple of issues. First, to calculate the mass of the wire, you would multiply the density times the volume of the wire. The mass you found using length^3 would be the mass of a cube of titanium 2.8 m on each side. But this wire is a skinny cylinder, and so it's volume is its length times its cross-sectional area.

But the mass of the wire is not what goes into the Young's modulus calculation. The idea is this: first the wire had a length of 2.8 m, then because somebody hung a 109 kg mass on it, it stretched by 1.41 cm. So the force that we use in calculating the stress is the force that makes the wire stretch, so here the mass needs to be 109 kg (because the force causing the stretch is the weight of that mass).

(The reason we don't need to include the weight of the wire at all is because whatever effect it had on the length was already accounted for in the original 2.8 m length. Only the extra mass caused the extra length.)
  • #5
That makes sense.

So instead of trying to figure out the wire's mass we would find the mass of the weight added to the end (though in this case it's given).

So it's simply (9.81m/s*109kg)/(pi*(7.5E-4)^2) = 6.0509E8 = stress

6.0509E8/5.0357E-3 = 1.2016E11 - this seems like a large number, though titanium is 105-120 GPa.

I think the units should be kg/(ms^2), though mine doesn't seem to work out that way.

1. What is Young's modulus?

Young's modulus, also known as the modulus of elasticity, is a measure of the stiffness or resistance to deformation of a material. It is defined as the ratio of stress (force per unit area) to strain (change in length per unit length) within the elastic range of a material.

2. Why is Young's modulus important?

Young's modulus is an important material property because it allows us to predict how a material will behave under different types of stress, such as tension or compression. It also helps in determining the maximum load a material can withstand without permanent deformation.

3. How is Young's modulus measured?

Young's modulus can be measured experimentally by subjecting a material to different levels of stress and measuring the corresponding strain. It can also be calculated using the material's stress-strain curve, which is obtained through tensile testing.

4. What factors affect Young's modulus?

The Young's modulus of a material is affected by various factors such as temperature, microstructure, and alloying elements. It also varies depending on the direction of applied stress and the presence of defects or imperfections in the material.

5. What is the Young's modulus for this alloy of titanium?

The Young's modulus for any specific alloy of titanium will vary depending on its composition and processing methods. It can range from 100 GPa to 120 GPa, with an average value of 110 GPa. This can also be affected by factors such as temperature and strain rate.

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