### Maximum hanging length Chain on Table with out slipping against friction

If a portion of chain is hanging from one face of table, due to its weight it tends to slip from the table. Up to certain length of hanging, it can stay on the table with out slipping.Chain is a body whose mass is uniformly distributed over its length. Mass of the chain is thus directly proportional to the mass of the chain as it is a one dimensional body.

Mass of the chain hanging can be determined as the product of length of the chain and linear density of the chain. Linear density is nothing but mass per unit length of the body. Thus we can identify the mass of the hanging part of the chain as well as mass of the portion of the chain that is lying on the surface of the table.

As some portion of chain is hanging down, its weight acts in downward direction and its it creates a tightness in the chain. This acts like the tension in the chain and in turn in acts on the horizontal portion of the chain that is on the table. This tension tend to slip the chain away from the table and opposite to it, frictional force is generated against the relative motion.

Frictional force here is also the product of coefficient of friction and normal reaction. Portion of the chain on the table applies its weight on the floor of the table and it acts like action. The chain applies reaction on the part of chain on the table and it is nothing but normal reaction. So we can calculate the frictional force acting.

For the chain not to slip, this frictional force has to be balanced by the tension in the chain. Tension in the chain is nothing but the weight of the portion of the chain hanging.

Using this equating forces , we can get the maximum length of chain that can be hung from table with out slipping.The expression is derived in the video as shown below.

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