A uniform chain of 6 m length is placed on a table such that a part of its length is hanging over the edge of the table. The system is at rest. The co-efficient of static friction between the chain and the surface of the table is 0.5, the maximum length of the chain hanging from the table is ____m.
To determine the maximum length of the chain that can hang over the edge of the table without slipping, we need to consider the balance of forces caused by static friction. The total weight of the chain is proportional to its length. Let \(L\) be the total length of the chain and \(x\) be the portion of the chain hanging off the table. We know:
Total chain length, \(L = 6 \text{ m}\).
Static friction co-efficient, \(\mu = 0.5\).
The weight of the overhanging part is \(\lambda x g\), where \(\lambda\) is the mass per unit length and \(g\) is acceleration due to gravity. The weight of the chain on the table is \(\lambda(L-x)g\). Static friction force, \(F_f\), is given by \(F_f = \mu \lambda(L-x) g\). For equilibrium:
\(\lambda x g = \mu \lambda(L-x) g\)
Upon solving:
\(x = 0.5(6-x)\)
Expanding:
\(x = 3 - 0.5x\)
Combining terms:
\(1.5x = 3\)
Solving for \(x\):
\(x = 2\) m
This is within the expected range of 2 to 2 m, confirming our solution is consistent with the problem's parameters. Therefore, the maximum length of the chain hanging over the table without slipping is 2 m.
