Physics Help (Solids and Elastic Moduli)?

Question

I am having trouble with this question, I know the answer is
5.4 *10^-5 but I don't know how to go about finding the answer:

A 45 kg traffic light is suspended form 2 steel cables of equal length and radii of 0.005 m. If each cable makes a 15 degree angel with the horizontal, what is the fractional increase in their length due to the weight of the light?

If someone can please help explain the proceedure to finding the solution, it would most apreciated

Answer 1

First you have to find the tension in each of the steel cables. Since each of them are at 15 degrees from the horizontal, they carry the same amount of tension. The weight of the traffic light is 45kg * gravity. Since this hangs straight down, the vertical components of the tension in the cables must equal this weight. Let's call the tension in each cable T. The vertical component is then T*sin(15 degrees). Since there are two cables, 2*T*sin(15 degrees) = 45kg * gravity. Then you can solve for T.

Next, solve for the stress in the cable. Stress is force / area. In this case, force and tension are the same and you can calculate the area from the radii given.

Finally, use Hooke's Law: Stress = Modulus * Strain. What the question really asked was to find the strain. So Strain = Stress / Modulus. I'll have to assume you can look up the modulus of steel somewhere, according to your book, but since you know the answer, you can also back calculate.

Ba-da-bing, ba-da-boom.