Problem 4.

(Physics for scientists and engineers - Serway and Jewett - Chapter 12 - Problem 31)

A walkway suspended across a hotel lobby is supported at numerous points along its edges by a vertical cable above each point and a vertical column underneath. The steel cable is 1.27 cm in diameter and is 5.75 m long before loading. The aluminum column is a hollow cylinder with an inside diameter of 16.14 cm, an outside diameter of 16.24 cm, and unloaded length of 3.25 m. When the walkway exerts a load force of 8500 N  on one of the support points, how much does the point move down?

Solution:

In this problem we need to use the Hook's law for the elastic force, expressed in terms of modulus of elasticity (Young's modulus):

where is an elastic modulus, is the cross-section area, is the length of the cable (column), and is the change of the length of the cable (column).

The walkway with the load is in equilibrium. It means that the net force acting on the walkway is zero. There are three forces acting on the walkway: the load force, the elastic force due to cable, and the elastic force due to column. The forces are shown in the figure. Then in equilibrium we have:

..................................................(1)

The elastic forces can be found from the Hook's law:

It is important that is the same in both expressions.

The cross-section areas are the following:

The elastic modulus can be found from the tables

Then equation (1) becomes

From this equation we can find x: