An 1567 kg car, being lifted at a steady speed by a crane, hangs at the end of a cable whos radius is .0084. The cable is 5m in length and stretches by .0081 becuase of the weight of the car. What is young's modulus for the cable?
Use this formula:
E*(delta L)/L = (F/Area)
E is the Young's modulus you want to solve for
F is the car's weight, in Newtons, M g
Area = pi* R^2
delta L = 0 0081 m
L = 5 m
To find Young's modulus (E) for the cable, we can use Hooke's law and the given information.
Hooke's law states that the stress (σ) in a material is directly proportional to the strain (ε) applied to it. Mathematically, this can be written as:
σ = E * ε
Where:
σ = Stress (measured in Pascal, Pa)
E = Young's modulus (measured in Pascal, Pa)
ε = Strain (unitless)
In this case, the strain is given as the change in length divided by the original length:
ε = ΔL / L
Given:
Original length (L) = 5 m
Change in length (ΔL) = 0.0081 m
Substituting the values into the equation, we can solve for the strain:
ε = 0.0081 / 5
ε ≈ 0.00162
To find the stress, we can use the weight of the car and the cross-sectional area of the cable.
The weight (F) of the car is given by:
F = m * g
Where:
m = Mass of the car = 1567 kg
g = Acceleration due to gravity ≈ 9.8 m/s^2
F = 1567 * 9.8
F ≈ 15331.6 N
The cross-sectional area (A) of the cable is given by:
A = π * r^2
Where:
r = Radius of the cable = 0.0084 m
A = π * 0.0084^2
A ≈ 0.000221 square meters
Now, we can find the stress (σ) using the formula:
σ = F / A
σ ≈ 15331.6 / 0.000221
σ ≈ 69399819.5 Pa
Finally, substituting the calculated values for stress (σ) and strain (ε) into Hooke's law, we can solve for Young's modulus (E):
E = σ / ε
E ≈ 69399819.5 / 0.00162
E ≈ 42804617284.8 Pa
Therefore, the Young's modulus for the cable is approximately 42.8 GPa (gigapascals).
To determine Young's modulus for the cable, we need to use the formula:
Young's modulus (Y) = stress / strain
The stress can be calculated by dividing the force applied to the cable by its cross-sectional area:
Stress (σ) = Force / Area
In this case, the force is the weight of the car, which can be calculated using the formula:
Force (F) = mass * acceleration due to gravity
The mass of the car is given as 1567 kg, and the acceleration due to gravity is approximately 9.8 m/s^2.
The area can be calculated using the formula for the area of a circle:
Area = π * radius^2
In this case, the radius is given as 0.0084 m.
Now, let's calculate the force applied to the cable:
Force (F) = 1567 kg * 9.8 m/s^2
Next, let's calculate the area of the cable:
Area = π * (0.0084 m)^2
Now, we can calculate the stress:
Stress (σ) = Force / Area
After calculating the stress, we need to calculate the strain. In this case, the strain is given as the ratio of the change in length to the original length of the cable:
Strain (ε) = change in length / original length
The change in length is given as 0.0081 m, and the original length is given as 5 m.
Finally, we can calculate Young's modulus using the formula:
Young's modulus (Y) = stress / strain
Plug in the values for stress and strain to find Young's modulus.
Note that Young's modulus is specific to the material of the cable and may vary for different materials.