A triangle made of beams

Isosceles triangle \(\triangle ABC\) is constructed from three steel beams, with a fourth beam holding corner \(A\) in a fixed position. Keeping the isosceles triangular formation intact, the equal lengths of beams \(AB\) and \(AC\) both increase with the same rate. Since beam \(BC\) is always kept at a fixed length of \(2\text{ m}\), this movement causes \(\angle A\) to shrink at a rate of \(\frac { \pi }{ 4 } \text{ rad/s }\).

As soon as \(\triangle ABC\) becomes a right triangle, beam \(BC\) instantly detaches with a constant velocity and proceeds to move on an icy surface, parallel to the Earth's surface. In meters from \(A\), find the distance, \(d\), that the beam rolls before stopping.

Details and Assumptions

  • Assuming that this hypothetical scenario takes place on Earth, take the acceleration due to gravity to be at \(10\text{ m }/ \text{ s }^{ 2 }\).
  • Assume that the coefficient of kinetic friction between steel and ice is \(\frac { { \pi }^{ 2 } }{ 960 }\).

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