Final answer:
To find the minimum cross-sectional area of a steel cable needed to suspend a 220 kg chandelier, calculate the force on the cable, adjust the steel's tensile strength for the safety factor, and divide the force by the adjusted tensile strength.
Step-by-step explanation:
The question is asking for the minimum cross-sectional area required of a vertical steel cable to safely support a suspended 220 kg chandelier, considering a safety factor of 8.0 and given the tensile strength of steel. To determine this, we first calculate the maximum force (tensile strength) that the cable can withstand, and then adjust for the safety factor. The force due to the weight of the chandelier is the mass (220 kg) multiplied by the acceleration due to gravity (9.81 m/s2).
Tensile strength (with safety factor) = Tensile strength of steel / safety factor = (5.0 × 108 N/m2) / 8.0 = 6.25 × 107 N/m2.
The tensile force the cable needs to withstand is Force = mass × acceleration due to gravity = 220 kg × 9.81 m/s2 = 2158.2 N.
The required cross-sectional area (A) is then calculated by dividing the force by the adjusted tensile strength:
A = Force / Tensile strength = 2158.2 N / 6.25 × 107 N/m2
By calculating the above expression we get the minimum cross-sectional area required for the steel cable to safely support the chandelier with the given factors.