- #1
netnomad
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- Homework Statement
- 2.22 An ornament for a courtyard at a World's Fair is to be made up of four identical, frictionless metal spheres, each weighing 2√6 ton-weight. The spheres are to be arranged as shown in Fig. 2-20, with three resting on a horizontal surface and touching each other; the fourth is to rest freely on the other three. The bottom three are kept from separating by spot welds at the points of contact with each other. Allowing for a factor of safety of 3, how much tension T must the spot welds withstand?
- Relevant Equations
- ##\theta## is the angle between the horizontal plane the center of the top ball. From the geometry of the tetrahedron, ##\tan \theta = \sqrt{2}##.
Relation between ##T## and the force acting through the centroid:
$$T = \frac{F_T}{\cos 30^\circ}$$
Factor of safety gives:
$$T_{\text{safe}} = 3T = \frac{3F_T}{\cos 30^\circ}$$
Relating ##F_c## and ##F_T##:
$$F_c = 2F_T$$
The total acting force ##F##:
$$F = 3F_c$$
Substituting ##F_c = \frac{1}{3}F##:
$$F_T = \frac{1}{6}F$$
From equilibrium and using the principle of virtual work by applying a small displacement along the centroid (moving the ball to the center):
$$F = W \frac{dh}{dc}=W \tan \theta$$
Substituting ##F = W \tan \theta## into ##F_T##:
$$F_T = \frac{1}{6}W \tan \theta$$
Substituting ##F_T## into ##T_{\text{safe}}##:
$$\begin{aligned}
T_{\text{safe}} &= \frac{3F_T}{\cos 30^\circ}\\[6pt]
&= \frac{3}{\cos 30^\circ} \cdot \frac{1}{6}W \tan \theta \\[10pt]
&= \frac{W \tan \theta}{2 \cos 30^\circ}
\end{aligned}$$
Substituting ##\tan \theta = \sqrt{2}## and ##W = 2\sqrt{6}##:
$$T_{\text{safe}} = \frac{2\sqrt{6} \cdot \sqrt{2}}{2 \cdot \frac{\sqrt{3}}{2}} = \frac{2\sqrt{12}}{\sqrt{3}} = 4$$
After applying a small displacement along the axis between the center of the bottom ball and the projected center of the fourth ball, I got 4 as the answer, which is twice the correct value. I’m assuming I didn’t account for the fact that the tension force acts on both sides, so I ended up calculating it twice. Even though this makes sense to me intuitively, I’m not sure how to properly incorporate it into an equation or how to think about the tension force in this setup. Thank you!