# A triangle made of beams

**Classical Mechanics**Level 3

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 \(\frac { { \pi }^{ 2 } }{ 960 }\).