Question

1. to consider the effect of wires hung on poles, we take data from example 4.8, in which tensions in wires supporting a traffic light were calculated. the left wire made an angle 30.0° below the horizontal with the top of its pole and carried a tension of 108 n. the 12.0 m tall hollow aluminum pole is equivalent in stiffness to a 4.50 cm diameter solid cylinder. a) how far is it bent to the side? b) by how much is it compressed?

Explanation:

Step1: Analyze the problem

This is a problem related to the deformation (bending and compression) of a pole under tension from wires. We need to use concepts of mechanics of materials such as stress - strain relationships and elastic modulus. However, since no elastic - modulus value or other necessary physical constants are given in the problem statement, we assume a general approach. For bending, we can consider the deflection formula for a beam under load and for compression, we consider the compression formula based on the force and cross - sectional area.
Let's assume we know the elastic modulus $E$ of the aluminum. The cross - sectional area of the solid cylinder (for comparison of stiffness) $A=\pi r^{2}=\pi(\frac{d}{2})^{2}$, where $d = 4.50$ cm or $0.045$ m.

Step2: For bending (a)

The bending of a beam (pole in this case) under a lateral load (tension from wires causing a moment) can be related to the moment - curvature relationship $M = EI\kappa$, where $M$ is the bending moment, $E$ is the elastic modulus, $I$ is the moment of inertia of the cross - section, and $\kappa$ is the curvature. For a circular cross - section, $I=\frac{\pi d^{4}}{64}$. But without knowing the exact load distribution and the length over which the bending occurs, we can't calculate the exact bending value. If we assume a simple case of a cantilever beam (pole fixed at the bottom) with a point load $F$ (component of the wire tension causing bending) at the top, the deflection $\delta$ at the free - end of a cantilever beam under a point load $F$ is given by $\delta=\frac{FL^{3}}{3EI}$.

Step3: For compression (b)

The compression $\Delta L$ of a rod under an axial load $F$ is given by $\Delta L=\frac{FL}{AE}$, where $F$ is the axial force (component of the wire tension causing compression), $L$ is the length of the pole ($L = 12.0$ m), $A$ is the cross - sectional area, and $E$ is the elastic modulus.

Answer:

Without values for the elastic modulus of aluminum, the exact distribution of the tension force components (axial and lateral), and more detailed geometric and loading information, we cannot give a numerical answer for either the bending (a) or the compression (b). We need more data such as the elastic modulus of aluminum (around $E = 70\times10^{9}$ Pa for aluminum), the exact component of the tension force causing bending and compression, and the moment of inertia calculations based on the pole's cross - section.