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My By book says that, for equilibrium Limiting Friction$(f_L)$$=$Weight of hanging chain. But Why??

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I'm not sure whether that should be called limiting friction.

Whether or not the frictional force is equal to the limiting case will depend on the coefficient of static friction between the the chain and the table surface and the weight of the part of the chain resting on the table.

The chain won't slide down only when

$$\text{weight of the hanging part of chain}\leq\mu_s(\text{weight of chain on the table})$$

But what you're confused about is static friction which is a self-adjusting force.

For a given weight of chain resting on the table there are many possible weights hanging down for which the chain won't slide down, all of which are less than or equal to the limiting friction force which is$$\mu_s(\text{weight of chain on the table})$$

In each of these cases the acting frictional force (static friction) will be equal to the weight of the hanging chain.

In equilibrium, we can say that $$\text{weight of hanging chain}=\text{acting frictional force(static friction)}\leq\mu_s\text{(weight of chain on the table)}$$

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