A 500 N weight approximately 100 lb hangs from a 100 m long

A 500 N weight (approximately 100 lb) hangs from a 1.00 m long steel wire as shown in the diagram. What must the diameter of the wire be such that the wire elongation does not exceed 1.00 mm? Young\'s modulus for steel is E= 200 times 10^9 N/m^2.

Solution

Suppose \'r\' be the radius of the steel wire.

So, crossection area, A = pi*r^2

Given the weight, W = 500 N

So, stress = W/A = W / (pi*r^2)

Further, the requisite elongation, delta l = 1.0x10^-3 m

and length of the wire, L = 1.0 m

So, strain = (delta l) / L = 10^-3

Now, Young\'s modulus, E = stress / strain

Putting the given values -

200x10^9 =  [W / (pi*r^2)] / 10^-3 = 500 / (10^-3xpi*r^2)

=> r^2 = 500 / (10^-3xpi*200x10^9) = 7.959 x 10^-7 = 0.7959 x 10^-6

=> r = 0.89213 x 10^-3 m = 0.89213 mm

Therefore, the required diameter of the wire, d = 2r = 1.784 mm