is placed in a horizontal slit between two tables, so that one of the side faces is vertical. How small can the width of the slit be made in before the prism falls out of the slit.
A right prism that has an equilateral triangular base with lengthThere is no friction between the prism and the tables and the prism is made of a homogeneous material.
The edges of the slit are parallel
This was a past IPHO problem.
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Since the whole system is balanced at the limiting case (when the prism just begins to topple),
Net vertical force is balanced: m g = N 1 c o s ( 3 0 o )
Net horizontal force is balanced: N 2 = N 1 s i n ( 3 0 o )
Thus, we obtain N 2 = 3 m g ....................................................................... (1)
Net torque about point A is also zero since the system is in rotational equilibrium.
m g ( 2 3 a − d ) = N 2 3 d ......................................................................... (2)
Combining (1) and (2) and solving,we get: d = 8 a 3 .
Plug in the values to get the answer as 0 . 1 0 8 2 5