The unending streak of circles and sticks

A large number of sticks (with mass density \(\rho\) per unit length) and circles (with radius \(R\)) lean on each other, as shown in the figure below.

Each stick makes an angle \(\theta\) with the horizontal and is tangent to a circle at its upper end. The sticks are hinged to the ground, and every other surface is friction less.

In the limit of a very large number of sticks and circles, what is the normal force between a stick and the circle it rests on, very far to the right? If you find result as \( S \) give the as \( [S] \) , where [.] represents the ceiling function.

Details and assumptions:

  • The last circle, i.e. the circle at infinity is leaning against a wall (which has only the significance of stopping the whole system from moving)
  • Take \(\rho = 1, R = 15, g = 10\text{ m/s}^2, \theta = 74^\circ \).
  • Every value is given is S.I. system.

Hint: As always, generalize the result for any \( N\) and then take the limit as \(N\) tends to infinity.


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