Scarf modelled as a pulley system / F=ma exercise

[laser]

Homework Statement

Consider a scarf draped over a table. Model it as two particles of mass m1 and m2 joined by a model string passing over the edges of the table modelled as a model pulley. Assume the masses are proportional to the corresponding lengths of the scarf, i.e. the scarf’s mass is uniformly distributed. If the coefficient of static friction between the scarf and the table surface is μ, what proportion of the scarf can hang over the table before the scarf slips off the table?

Relevant Equations

F=ma

My teacher gave the above answer as a solution. However, I am not convinced that the proportion is really $$\frac{m_1}{m_1+m2}$$. If m2 << m1the proportion would be really big, right? But intuition tells me that it should be the opposite. Furthermore, if m2 >> m1, then one would expect the proportion to be "big". But it's the opposite :/.

What am I missing here?

 

[Orodruin]

The ratio $m_1/(m_1+m_2)$ is by definition the proportion of the scarf hanging over the edge. Since $m_1$ is the mass hanging over the edge, this should be expected to be large whenever $m_1 \gg m_2$.

 

[haruspex]

If m2 << m1the proportion would be really big, right? But intuition tells me that it should be the opposite. Furthermore, if m2 >> m1, then one would expect the proportion to be "big". But it's the opposite :/.

Intuition is telling you the same thing for both m2>>m1 and m2<<m1, yet is wrong both times? That does not sound possible.
Besides, it can never be "really big". It clearly cannot exceed 1.

 

[Orodruin]

Besides, it can never be "really big". It clearly cannot exceed 1.

I’d say 1 is a “really big” fraction of the scarf … it is all of it …