A steel wire is used to lift heavy object. The cable has a diameter of 14.25 mm and an initial length of 33.5 m. If the cable stretches 2.50 mm, what is the mass of the heavy object? Use 20.0 x 10^10 Pa for Young’s modulus for steel. Give your answer in kg and with 3 significant figures.
To find the mass of the heavy object, we need to calculate the change in length of the steel wire first.
1. Calculate the change in length using the formula:
ΔL = F * L0 / (A * Y)
where ΔL is the change in length, F is the force applied, L0 is the initial length, A is the cross-sectional area, and Y is the Young's modulus.
2. Rearrange the formula to solve for force (F):
F = ΔL * A * Y / L0
3. Substitute the given values into the formula:
ΔL = 2.50 mm = 2.50 x 10^-3 m
L0 = 33.5 m
A = π * (d/2)^2, where d is the diameter of the wire
Y = 20.0 x 10^10 Pa
4. Calculate the cross-sectional area:
A = π * (14.25 mm / 2)^2
A = π * (0.01425 m)^2
A ≈ 0.000160 cm^2
5. Calculate the force (F):
F = (2.50 x 10^-3 m) * (0.000160 cm^2) * (20.0 x 10^10 Pa) / 33.5 m
F ≈ 2.38 x 10^7 N
6. Lastly, calculate the mass of the heavy object using the formula:
F = m * g, where m is the mass and g is the acceleration due to gravity.
Rearrange the formula to solve for mass:
m = F / g
7. Substitute the value of F and g into the formula:
m = (2.38 x 10^7 N) / (9.81 m/s^2)
m ≈ 2.43 x 10^6 kg
Therefore, the mass of the heavy object is approximately 2.43 x 10^6 kg.
To find the mass of the heavy object, we need to calculate the change in length of the steel wire and then use it to find the force exerted on the wire. From there, we can use the force to calculate the mass of the object.
1. Calculate the change in length:
The change in length of the wire is given as 2.50 mm. However, we need to convert it to meters to be consistent with other units. So, the change in length (ΔL) is 2.50 mm ÷ 1000 = 0.0025 m.
2. Calculate the force exerted on the wire:
The force (F) exerted on the wire is given by Hooke's Law, which states that the force is directly proportional to the change in length of the wire. The equation is:
F = (Y * A * ΔL) / L
where:
F = force exerted on the wire
Y = Young's modulus for steel (20.0 x 10^10 Pa)
A = cross-sectional area of the wire
ΔL = change in length of the wire (0.0025 m)
L = initial length of the wire (33.5 m)
We need to calculate the cross-sectional area of the wire before proceeding with the force calculation.
3. Calculate the cross-sectional area:
The cross-sectional area (A) of the wire is given by the equation:
A = π * (r^2)
where:
A = cross-sectional area
π = 3.14159 (approximately)
r = radius of the wire (half of its diameter)
The diameter of the wire is 14.25 mm, so the radius is 14.25 mm ÷ 2 = 7.125 mm = 0.007125 m.
Substituting the values into the equation, we find:
A = 3.14159 * (0.007125)^2
Calculating this, we get A ≈ 0.000159649 m^2.
4. Calculate the force exerted on the wire:
Now that we have the cross-sectional area, we can calculate the force exerted on the wire using Hooke's Law:
F = (Y * A * ΔL) / L
F = (20.0 x 10^10 Pa * 0.000159649 m^2 * 0.0025 m) / 33.5 m
Calculating this, we find F ≈ 0.0029781 N.
5. Calculate the mass of the heavy object:
Lastly, we can calculate the mass of the heavy object using the formula:
F = m * g
where:
F = force exerted on the wire (0.0029781 N)
m = mass of the heavy object
g = acceleration due to gravity (9.8 m/s^2)
Rearranging the formula, we get:
m = F / g
Substituting the values, we find:
m = 0.0029781 N / 9.8 m/s^2
Calculating this, we find m ≈ 0.000304 kg.
Therefore, the mass of the heavy object is approximately 0.000304 kg with 3 significant figures.