Chapter 11.2, Problem 65E

To determine

To find: The tension in each of the supporting cables shown in the figure.

Answer to Problem 65E

The tension in each of the supporting cables 10.98 .

Explanation of Solution

Given information: The weight of a crate is 500 newton . Figure is given below.

  

Calculation:

The forces on the wires are equal due to symmetry .Also since the wires are equally spaced that implies that they will form ${60}^{\circ }$ angles on the circular disc. We consider the center of the disc as the origin of the three dimensional coordinate system.

The support point ,P has zero x,y coordinates and a z coordinate obtained using the Pythagorean Theorem. $z=\sqrt{{60}^2-{24}^2}=55.$ The coordinates for the support point are P=(0,0,55).

The downward force on the disc is w=<0,0,-30>.

Choosing the rightmost point, A and knowing that the angle on the disc is ${120}^{\circ }$ we get from basic trigonometry that the triangle formed on the disc are isosceles and that $x=r\cos {30}^{\circ }=20.78$ and $y=r\sin {30}^{\circ }=12$ . So the coordinates for point A given the coordinate system are A=(-20.78,-12.0).

The equilibrium equation gives us $u+v+z=-w\Rightarrow 3u=-w\Rightarrow u=-\frac{1}{3}w$ .

We also have that $u=\Vert u\Vert \frac{\overrightarrow{AP}}{\Vert \overrightarrow{AP}\Vert }=\Vert u\Vert <0.34,0.2,0.91>.$

We finally get that $\Vert u\Vert 0.91=10\Rightarrow \Vert u\Vert =\mathrm{10.98.}$