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.
Net force down = static friction force ,because there will be a balance of tension forces at the edge.
moff*g = (mtot-moff)*g*(0.395)
(mtot/moff) -1 = 1/0.395 = 1.532
mtot/moff = 2.532
moff/mtot = 0.395
0.395
that is wrong
To determine the maximum fraction \([m_{\text{off}}/m_{\text{total}}]\) that can hang over the edge without causing the whole washcloth to slide off the table, we can analyze the forces acting on the washcloth.
1. Consider the forces on the washcloth:
- The gravitational force acting downward on the washcloth is \(F_{\text{gravity}} = m_{\text{total}} \cdot g\), where \(m_{\text{total}}\) is the total mass of the washcloth and \(g\) is the acceleration due to gravity.
- The normal force exerted upward by the table on the washcloth is equal to the gravitational force: \(F_{\text{normal}} = m_{\text{total}} \cdot g\).
- The frictional force \(F_{\text{friction}}\) keeps the washcloth from sliding.
2. To determine the maximum fraction, we need to find the point at which the frictional force is at its maximum. The maximum frictional force is given by \(F_{\text{friction,max}} = \mu_s \cdot F_{\text{normal}}\), where \(\mu_s\) is the coefficient of static friction.
3. Substitute the values into the equation: \(F_{\text{friction,max}} = \mu_s \cdot m_{\text{total}} \cdot g\).
4. Since the frictional force opposes the gravitational force, it must be equal to or greater than the gravitational force in order to prevent the washcloth from sliding. Thus, we can set \(F_{\text{friction,max}}\) equal to \(F_{\text{gravity}}\) and solve for \(m_{\text{off}}\).
\(F_{\text{friction,max}} = F_{\text{gravity}}\)
\(\mu_s \cdot m_{\text{total}} \cdot g = m_{\text{total}} \cdot g\)
\(\mu_s \cdot m_{\text{total}} = m_{\text{total}}\)
\(\mu_s = \frac{m_{\text{off}}}{m_{\text{total}}}\)
5. Rearrange the equation to solve for \(\frac{m_{\text{off}}}{m_{\text{total}}}\):
\(\frac{m_{\text{off}}}{m_{\text{total}}} = \mu_s\)
So, the maximum fraction \(\frac{m_{\text{off}}}{m_{\text{total}}}\) that can hang over the edge without causing the whole washcloth to slide off the table is equal to the coefficient of static friction \(\mu_s\). In this case, the maximum fraction is 0.395.