Tension for the crown

The blacksmiths of the kingdom of Mechania are wondering how much tension is holding a crown together as it rests on the head of their sovereign. The king of Mechania has a frictionless, perfectly spherical head with radius \(r = 10 \text{ cm}\), and the crown is a thin cord with a length of \( \ell = 37.7 \text{ cm}\) and a mass of \(m = 628.32 \text{ g}.\)

Compute the tension in the crown (in Newtons).

Assumptions:

  • The crown has no thickness and rests perfectly level (that is, it makes a circle which is contained in a plane parallel to the horizontal).
  • The gravitational constant of Mechania is \(g = 10 \text{ m/s}^2. \)
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