A damp washcloth is hung over the edge of a table to dry. Thus, part (mass = mon) of the washcloth rests on the table and part (mass = moff) does not. The coefficient of static friction between the table and the washcloth is 0.395. Determine the maximum fraction [moff/mtotal] that can hang over the edge without causing the whole washcloth to slide off the table
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To determine the maximum fraction of the washcloth that can hang over the edge without causing it to slide off the table, we need to consider the forces acting on the washcloth.
The weight of the washcloth can be divided into two parts: the portion resting on the table (mg * mon) and the portion hanging over the edge (mg * moff).
Now, let's analyze the forces. The force of gravity acting on the washcloth is the weight (mg), which acts vertically downwards. The maximum frictional force (Ff) that can be exerted by the table on the washcloth without causing it to slide off is given by the coefficient of static friction (µs) multiplied by the normal force.
The normal force is the force exerted by the table to support the weight of the washcloth. It is equal to the sum of the forces exerted on the washcloth, which is the combination of the portion on the table (mg * mon) and the portion hanging over the edge (mg * moff).
So, the normal force (N) is given by:
N = (mg * mon) + (mg * moff)
The maximum frictional force (Ff) is given by:
Ff = µs * N
For the washcloth to remain in equilibrium without sliding off, the maximum frictional force (Ff) exerted by the table must be equal to the weight of the portion hanging over the edge (mg * moff). Therefore, we have:
Ff = mg * moff
Setting these two equations equal, we can solve for the maximum fraction (moff/mtotal) of the washcloth that can hang over the edge without sliding off:
mg * moff = µs * [(mg * mon) + (mg * moff)]
Dividing both sides by mg, we get:
moff = µs * (mon + moff)
Rearranging the equation, we get:
moff - (µs * moff) = µs * mon
Factoring out moff, we have:
moff * (1 - µs) = µs * mon
Finally, dividing by (1 - µs), we get the expression for the maximum fraction:
moff/mtotal = µs * mon / (1 - µs)
Substituting the given coefficient of static friction (µs = 0.395) into the equation will give you the maximum fraction (moff/mtotal) that can hang over the edge without causing the whole washcloth to slide off the table.