Question

A metal wire 750cm3 long and 0.13cm3 in diameter stretches 0.350cm3 when a load of 800g is hung on its end . Find the stress , strain , percentage and Young’s modulus for the material of the wire

2. A tank used for filling Argon balloons has a volume of 0.450mL3 and contains 2.00 ml of Argon gas at 20.0K . Assuming that the helium behaves like an ideal gas
a. What is the total transitional kinetic energy of the molecules of the gas .
b. What is the average kinetic energy per molecule ?

Answer 1

We can use the formula for stress to find the stress in the wire:

stress = force/area

where force is the weight of the load and area is the cross-sectional area of the wire.

The weight of the load is 800g, which is 0.8kg. The cross-sectional area of the wire is given by:

area = πr^2 = π(0.065cm)^2 = 0.01326cm^2

We convert the length of the wire to meters and the diameter to meters as well:

length = 750cm = 7.5m

diameter = 0.13cm = 0.0013m

The strain is given by:

strain = change in length/original length

We can find the change in length from the given information:

change in length = 0.350cm = 0.0035m

The original length of the wire is:

original length = length / (πd^2/4) = 7.5m / (π(0.0013m)^2/4) = 4.525m

Using the formula for Young's modulus:

Young's modulus = stress/strain

We can now calculate the stress, strain, and Young's modulus:

stress = force/area = 0.8kg * 9.81m/s^2 / 0.01326cm^2 = 46,016 Pa

strain = change in length/original length = 0.0035m/4.525m = 0.000773

Young's modulus = stress/strain = 46,016 Pa / 0.000773 = 59,521,438 Pa

The percentage strain is the strain multiplied by 100:

percentage strain = strain * 100 = 0.000773 * 100 = 0.0773%

a. The total transitional kinetic energy of the molecules of an ideal gas is given by the formula:

total kinetic energy = (3/2) * n * R

Answer 2

1. To find stress, we use the formula:
Stress = Force / Area

We are given the length and diameter of the wire, so we can find its cross-sectional area:
Area = πr² = π(0.065cm)² = 0.0133cm²

We are also given the load (force) and the amount of stretching, so we can find the stress:
Stress = Force / Area = (800g)(9.8m/s²) / (0.0133cm²) = 4,630,000 Pa

To find strain, we use the formula:
Strain = Change in length / Original length

We are given the amount of stretching and the original length of the wire, so we can find the strain:
Strain = Change in length / Original length = 0.350cm / 750cm = 0.000467

To find the percentage of strain, we multiply the strain by 100:
Percentage of strain = Strain x 100 = 0.0467%

To find Young's modulus, we use the formula:
Young's modulus = Stress / Strain

We already found the stress and strain, so we can plug them in to find Young's modulus:
Young's modulus = Stress / Strain = 4,630,000 Pa / 0.000467 = 9.91 x 10^9 Pa

2. a. To find the total translational kinetic energy of the molecules of the gas, we use the formula:
Total kinetic energy = (3/2) x n x R x T

where n is the number of moles of gas, R is the universal gas constant (8.31 J/mol-K), and T is the temperature in Kelvin.

We are given the volume of the tank and the amount of gas it contains, so we can find the number of moles of gas:
n = PV / RT = (2.00 mL) / (8.31 J/mol-K x 20.0 K x 0.000450 L) = 0.0536 mol

Now we can plug in the values to find the total kinetic energy:
Total kinetic energy = (3/2) x n x R x T = (3/2) x 0.0536 mol x 8.31 J/mol-K x 20.0 K = 12.6 J

b. To find the average kinetic energy per molecule, we use the formula:
Average kinetic energy per molecule = (3/2) x R x T

We already know the temperature and universal gas constant, so we can plug them in to find the average kinetic energy per molecule:
Average kinetic energy per molecule = (3/2) x R x T = (3/2) x 8.31 J/mol-K x 20.0 K = 249 J/mol

To convert from Joules per mole to Joules per molecule, we divide by Avogadro's number (6.02 x 10^23 molecules/mol):
Average kinetic energy per molecule = 249 J/mol / (6.02 x 10^23 molecules/mol) = 4.14 x 10^-22 J/molecule