Tipping point

A right prism that has an equilateral triangular base with length a = 50 c m a=50cm is placed in a horizontal slit between two tables, so that one of the side faces is vertical. How small can the width d d of the slit be made in m e t e r s meters before the prism falls out of the slit.

There 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.


The answer is 0.108.

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

Maharnab Mitra
Jun 3, 2014

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 ) mg= N_1 cos(30^o)

Net horizontal force is balanced: N 2 = N 1 s i n ( 3 0 o ) N_2 = N_1 sin(30^o)

Thus, we obtain N 2 = m g 3 N_2= \frac{mg}{\sqrt{3}} ....................................................................... (1)

Net torque about point A is also zero since the system is in rotational equilibrium.

m g ( a 2 3 d ) = N 2 d 3 mg (\frac{a}{2 \sqrt{3}}-d) = N_2 \frac{d}{\sqrt{3}} ......................................................................... (2)

Combining (1) and (2) and solving,we get: d = a 3 8 \boxed{d = \frac{a \sqrt{3}}{8}} .

Plug in the values to get the answer as 0.10825 \boxed{0.10825}

i want to cry,my answer is wrong,because i put the a with 60 cm :'(.....im so sloppy...any advices guys?to avoid a silly mistake

Christian Leonardi - 6 years, 4 months ago

NIce!Did you painstakingly draw this?I guess there is only one way to do this,did it the same way.

Arif Ahmed - 6 years, 7 months ago

The problem asks when the prism "the prism falls out of the slit". This solution is the point where the prism starts to rotate and thus no longer flush with surface B. But at that point the prism does not fall out of the slit!

Kevin Lehmann - 3 years, 10 months ago
Beakal Tiliksew
May 27, 2014

Imgur Imgur

Here is a solution , more visual

Cool problem!

Steven Zheng - 6 years, 9 months ago

The solution should be in cm so 10.8 instead of 0.108

Ahmed Salah - 7 years ago

Log in to reply

answer should be in metres

Anurag Shekhar - 7 years ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...