. A platform with boat weighs 10,000 N. Four cables are attached to the platform as shown; the edges of the platform are always a horizontal distance of 2 m from the pulleys. As the platform rises smoothly and evenly, the length L of the cables decreases and the angle θ gets smaller.
The pulleys are 5 m above the waterline. The cables have a diameter of 5 mm, and are made of a material with mechanical properties E = 200 GPa, σy = 382 MPa, and ν = 0.27. Assume a factor of safety of 1.5.
a) How high will the boat/platform be when the tension in the cables exceeds the allowable stress?
b) At that height, determine the length L of a cable from the platform to its pulley. How much will that cable length L deform due to the tension at that height?
c) What will be the final diameter of the cable?
To solve this problem, we need to consider the forces acting on the platform and the cables, as well as the material properties of the cables.
a) To find the height at which the tension in the cables exceeds the allowable stress, we need to calculate the maximum tension in the cables. The tension in each cable is equal to the weight of the platform divided by the number of cables (4).
Tension = Weight of platform / Number of cables
Tension = 10,000 N / 4
Tension = 2,500 N
Next, we need to determine the cross-sectional area of the cables. The diameter of the cables is given as 5 mm, so the radius (r) is half of that, which is 2.5 mm.
Area of cable = π * r^2
Area of cable = π * (2.5 * 10^-3)^2
Area of cable = 19.63 * 10^-6 m^2
Now we can calculate the stress in the cables using the formula:
Stress = Tension / Cross-sectional area
Stress = 2,500 N / 19.63 * 10^-6 m^2
Stress = 1.27 * 10^8 N/m^2
Since the factor of safety is 1.5, we multiply the stress by the factor of safety to obtain the allowable stress:
Allowable stress = Stress * Factor of safety
Allowable stress = 1.27 * 10^8 N/m^2 * 1.5
Allowable stress = 1.905 * 10^8 N/m^2
To find the height where the tension exceeds the allowable stress, we need to find the length L of the cables at that height. We can use the relationship between length L, angle θ, and the horizontal distance between the pulleys and the platform.
L = 2 m / cos(θ)
Since we are looking for the height where the tension exceeds the allowable stress, we need to rearrange the equation to solve for cos(θ).
cos(θ) = 2 m / L
Now we can substitute the value of allowable stress into the stress equation and solve for the height:
1.905 * 10^8 N/m^2 = Tension / Area of cable
Tension = Stress * Area of cable
Tension = 1.905 * 10^8 N/m^2 * 19.63 * 10^-6 m^2
Tension = 3,740 N
cos(θ) = 2 m / L
cos(θ) = 2 m / (Length of cable)
L = 2 m / cos(θ)
Length of cable = 2 m / cos(θ)
Now we can find the height by solving the equation:
Tension = Weight of platform / Number of cables
3,740 N = 10,000 N / 4
3,740 N = 2,500 N
2,500 N = 10,000 N / 4
L = 2 m / cos(θ)
Length of cable = 2 m / cos(θ)
Substituting the values:
Length of cable = 2 m / cos(θ)
Length of cable = 2 m / cos(θ)
Length of cable = 2 m / 2
Length of cable = 1 m
Therefore, the platform will be 1 meter high when the tension in the cables exceeds the allowable stress.
b) Now, we need to determine how much the cable will deform due to the tension at that height. To do this, we need to calculate the strain in the cable. Strain is defined as the change in length divided by the original length.
Strain = Change in length / Original length
The change in length is given by:
Change in length = Tension * Length of cable / (Area of cable * Young's modulus)
Substituting the values:
Change in length = 3,740 N * 1 m / (19.63 * 10^-6 m^2 * 200 * 10^9 N/m^2)
Change in length = 0.0095 m
Therefore, the cable will deform by 0.0095 meters at that height.
c) Finally, we need to determine the final diameter of the cable. We can calculate the final diameter using the formula for the change in length in terms of the change in diameter:
Change in length = (Change in diameter) * Length of cable * Poisson's ratio
Rearranging the equation to solve for the change in diameter:
Change in diameter = Change in length / (Length of cable * Poisson's ratio)
Substituting the values:
Change in diameter = 0.0095 m / (1 m * 0.27)
Change in diameter = 0.034 m
The final diameter of the cable will be the original diameter plus the change in diameter:
Final diameter = Original diameter + Change in diameter
Substituting the values:
Final diameter = 5 mm + 0.034 m
Final diameter = 5.034 mm
Therefore, the final diameter of the cable will be 5.034 mm.