A circular rope of weight W and radius r = 2 R is resting on a smooth sphere of radius R. The tension in rope is

Question

A circular rope of weight W and radius r = 2 R ​ is resting on a smooth sphere of radius R. The tension in rope is

TestHub · Accepted Answer

Take a small element of the rope. The sphere pushes it with a normal force $F$ directed along the radius; the rope tension acts along the rope. For the whole rope (radius $r = R /2$ , so the rope sits at polar angle $θ$ with $sin θ = r / R = \frac{1}{2} \Rightarrow θ = 3 0^{\circ}$ ).

Vertical balance: $F cos θ = W$ . Horizontal (radial, towards the axis of the rope): $F sin θ = T \cdot 2 π$ .

$tan θ = \frac{2 π T}{W} \Rightarrow T = \frac{W tan 3 0 ^{\circ}}{2 π} = \frac{W}{2 π 3} = \frac{W}{π 12}$ .

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