Question

The length of a chain is 1 m. The chain is lying on a
horizontal table. The coefficient of friction between the
chain and the table is 0.25. What is the maximum
length of the chain that can hang out without slipping?
[20 cm)​

Answer 1

Solution :-

As per the given data ,

• Length of chain (L)= 1 m

• Coefficient of friction (μ) = 0.25

Let , the length of the hanging part of the chain be x

Length of remaining part of the chain = L - x

we know that ,

➜ m = λ x l

here ,

• λ = linear mass density

The mass of hanging part ,

➜ m = λ x

The mass of the remaining part of the chain ,

➜ m ' = λ(L - x )

The frictional force provided by the table prevents the chain from slipping . For the chain to hang without slipping both the forces that the frictional force ( acting on remaining part)and the gravitational force( acting on the hanging part ) must be equal

Hence ,

➜ f = Fg  

➜ μN = mg

➜ μλ(L - x )g = λxg

➜ μ ( L - x) = x

➜ μL - μx = x

➜ μL = x + μx

➜ μL = x (1+ μ)

➜ x = μL / ( 1 + μ)

➜ x = 0.25 x 1 / 1.25

➜ x = 1 / 5

➜ x = 0.2 m

➜ x = 20 cm

The maximum length of the chain that can hang out without slipping is 20 cm