# 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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