A hanging wire made out of titanium with diameter 0.080 cm is initially 2.3 m long. When a 78 kg mass is hung from it, the wire stretches an amount 2.89 cm. A mole of titanium has a mass of 47.9 grams, and its density is 4.51 g/cm3. Based on these experimental measurements, what is the Young's modulus for titanium?

Find the effective spring stiffness of one interatomic bond in titanium.

Well, if you're looking for the effective spring stiffness of one interatomic bond in titanium, I must say you're really digging into the nitty-gritty of things! You're like a detective looking for clues in the atomic underworld!

Anyway, let's get cracking. To find the Young's modulus for titanium, we need to use Hooke's Law, which states that the strain (change in length) of a material is directly proportional to the stress (force applied) on it.

So, the strain (ε) can be calculated as the change in length divided by the original length:

ε = ΔL / L₀

where ΔL is the change in length (2.89 cm) and L₀ is the original length (2.3 m). But wait, we need to convert everything to the same units, so let's convert the length to centimeters first:

L₀ = 2.3 m * 100 cm/m = 230 cm

Now, we can plug in the values and solve for ε:

ε = 2.89 cm / 230 cm ≈ 0.0126

Next, we need to find the stress (σ), which is the force applied (F) divided by the cross-sectional area (A) of the wire:

σ = F / A

But we don't know the force, right? Well, we can calculate it using the weight of the mass (force due to gravity):

F = m * g

where m is the mass (78 kg) and g is the acceleration due to gravity (approximately 9.81 m/s²). We know the density of titanium (4.51 g/cm³) and the diameter (0.080 cm) of the wire, so we can find its cross-sectional area (A):

A = π * (d/2)²

where d is the diameter (0.080 cm). Let's compute it:

A = π * (0.080 cm / 2)² ≈ 0.00502 cm²

Now, let's calculate the force (F):

F = (78 kg) * (9.81 m/s²) ≈ 764 N

Finally, we can solve for the stress (σ):

σ = 764 N / 0.00502 cm² ≈ 152,390 N/cm²

Now we have both the strain (ε) and the stress (σ). Young's modulus (E) can be calculated by dividing stress by strain:

E = σ / ε

Let's do the math:

E = 152,390 N/cm² / 0.0126 ≈ 12,099,206 N/cm²

And there you have it! The Young's modulus for titanium is approximately 12,099,206 N/cm².

To find the Young's modulus for titanium, we can use the formula:

Young's modulus (Y) = stress / strain

First, let's find the stress:

Stress = Force / Area

The force is the weight of the 78 kg mass, which can be calculated as:

Force = mass * gravitational acceleration

Where gravitational acceleration is approximately 9.8 m/s^2 (assuming Earth's gravity). Hence,

Force = 78 kg * 9.8 m/s^2

Next, we need to find the cross-sectional area of the wire. The wire's diameter is given as 0.080 cm, so its radius (r) can be calculated as:

r = diameter / 2 = 0.080 cm / 2 = 0.040 cm = 0.0004 m

The cross-sectional area is then given by:

Area = π * r^2

Area = π * (0.0004 m)^2

Now, let's calculate the stress:

Stress = Force / Area

Finally, let's find the strain:

Strain = (Change in length) / (Original length)

Change in length = 2.89 cm = 0.0289 m (convert to meters)

Original length = 2.3 m

Now, let's find the strain:

Strain = 0.0289 m / 2.3 m

With the stress and strain calculated, we can now find the Young's modulus:

Young's modulus (Y) = stress / strain

Plug in the values:

Young's modulus (Y) = (Stress) / (Strain)

This will give you the Young's modulus for titanium.

To find the effective spring stiffness of one interatomic bond in titanium, we can use Hooke's law, which states that the force required to stretch or compress a spring is directly proportional to the displacement:

F = k * x

Where F is the force applied, k is the spring constant, and x is the displacement.

By rearranging the formula:

k = F / x

In our case, the force applied is the weight, calculated as:

Force = mass * gravitational acceleration

And the displacement (x) is the amount the wire stretches, which is given as 2.89 cm (or 0.0289 m). Hence,

k = (mass * gravitational acceleration) / (0.0289 m)

This will give you the effective spring stiffness of one interatomic bond in titanium.

To find the Young's modulus for titanium, we can use the formula:

Young's modulus (E) = (stress) / (strain)

where stress is the force applied divided by the cross-sectional area, and strain is the change in length divided by the original length.

1. Calculate the stress:
Since the force applied is the weight of the mass, we can find it using:
Force = mass * acceleration due to gravity
Force = 78 kg * 9.8 m/s^2 = 764.4 N

The cross-sectional area can be calculated using the formula:
Area = π * (diameter/2)^2
Area = π * (0.080 cm / 2)^2
Area = 0.00502 cm^2 or 0.000502 m^2

Now, we can calculate the stress:
Stress = Force / Area
Stress = 764.4 N / 0.000502 m^2
Stress = 1,522,310 Pa

2. Calculate the strain:
Strain = (Change in length) / (Original length)
Strain = 2.89 cm / 230 cm
Strain = 0.0126

3. Calculate the Young's modulus:
Young's modulus (E) = Stress / Strain
Young's modulus (E) = 1,522,310 Pa / 0.0126
Young's modulus (E) ≈ 120,845,238 Pa or 120.8 GPa

Therefore, the Young's modulus for titanium is approximately 120.8 GPa.

To find the effective spring stiffness of one interatomic bond in titanium, we need to convert the Young's modulus to the spring constant for a single bond.

Spring constant (k) of a single bond = Young's modulus (E) / Average bond length

The bond length of titanium is not provided in the question, so we cannot calculate the spring constant without knowing it.

Young's modulus = (tensile stress)/(strain)

In this case the strain is
(delta L)/L = 2.89cm/230 cm
= 1.257*10^-2

and the stress is

sigma = (78*9.8 N)/[(pi/4)*(8^10^-4m)^2] = 764.4 N/5.027*10^-7 m^2)
= 1.52*10^9 N/m^2

So E = 1.21*10^11 N/m^2
= 121 GPa

A spring stiffness for an individual Ti-Ti interatomic pair can be estimated by dividing
(Tensile force)/(area occupied by one molecule)
by (stretch per intermolecular molecule pair).

You will need a characteristic intermolecular distance or diameter for Ti atoms in the solid. Call it d. Get that from the number density of Ti atoms, n.
d = n^(-1/3)

n = [4.51 g/cm^3/(47.9g/mole)]*6.02*10^23 atom/mole = 5.56*10^22 atom/cm^3
n^-1/3 = d = 2.6*10^-8 cm
= 2.6*10^-10 m
spring stiffness = k
=(Tension/d^2)/(strain*d)
= E/d = 1.21*10^11/2.6*10^-10
= 4.6*10^20 N/m