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 Newton. The spheres are to be arranged as shown, with three resti

Question

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 ​ Newton. The spheres are to be arranged as shown, 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, find the tension (in N) which each spot weld must be able to withstand.

TestHub · Accepted Answer

The line joining the top sphere's centre to a bottom centre makes angle $α$ with the vertical, where $cos α = 2/3$

Vertical balance of the top sphere: $3 N cos α = 26$ , giving $N = 2$ N

Each bottom sphere is pushed outward horizontally: $N sin α = \frac{2}{3}$ N

The two welds on each sphere balance this: $2 T cos 3 0^{\circ} = N sin α$ , so $T = \frac{2}{3}$ N

With safety factor 3, each weld must withstand $3 \times \frac{2}{3} = 2$ N

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