A 15-kg chandelier is suspended from the ceiling by four vertical steel wires. Each wire has an unloaded length of 5 m and a diameter of 6 mm, and each bears an equal load. When the chandelier is hung, how far do the wires stretch? (The Young's modulus for steel is 2.00 1011 N/m2.)
To find how far the wires stretch when the chandelier is hung, we can use Hooke's law, which states that the extension or stretch of an object is directly proportional to the force applied to it.
First, let's find the force on each wire.
Since the chandelier weighs 15 kg and the acceleration due to gravity is approximately 9.8 m/s^2, the force acting downwards on the chandelier is:
Force = mass x acceleration = 15 kg x 9.8 m/s^2 = 147 N
Since the load is distributed evenly among the four wires, the force acting on each wire is:
Force per wire = 147 N / 4 = 36.75 N
Now, let's calculate the cross-sectional area of one wire:
The diameter of the wire is given as 6 mm, which is equivalent to 0.006 m.
The cross-sectional area of a wire can be calculated using the formula:
Area = π * (diameter/2)^2
Substituting the values, we get:
Area = π * (0.006/2)^2 = π * (0.003)^2 = 0.00002827 m^2
Next, let's calculate the change in length or stretch of each wire.
Hooke's law states that the force is proportional to the stretch or extension of the wire:
Force = Young's Modulus * (change in length / original length) * Area
Rearranging the equation, we can find the change in length:
Change in length = (Force * original length) / (Young's Modulus * Area)
Substituting the values, we have:
Change in length = (36.75 N * 5 m) / (2.00 x 10^11 N/m^2 * 0.00002827 m^2)
Simplifying the equation, we find the change in length or stretch for each wire:
Change in length = 0.065 m
Therefore, each wire stretches by approximately 0.065 meters when the chandelier is hung.