Olympiad problems

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2013 IPhOO, 5

A uniform ladder of mass $m$ and length $\mathcal{L}$ is resting on a wall. A man of mass $m$ climbs up the ladder and is in perfect equilibrium with the ladder when he is $\frac{2}{3}\mathcal{L}$ the way up the ladder. The ladder makes an angle of $ \theta = 30^\circ $ with the horizontal floor. If the coefficient of static friction between the ladder and the wall is the same as that between the ladder and the floor, which is $\mu$, what is $\mu$, expressed to the nearest thousandth? [i](Proposed by Ahaan Rungta)[/i]

2017 F = ma, 6

Tags: torque

6) In the mobile below, the two cross beams and the seven supporting strings are all massless. The hanging objects are $M_1 = 400 g$, $M_2 = 200 g$, and $M_4 = 500 g$. What is the value of $M_3$ for the system to be in static equilibrium? A) 300 g B) 400 g C) 500 g D) 600 g E) 700 g

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