Question

A partly hanging uniform chain of length L is resting on a rough horizontal table. ℓ is the maximum possible length that can hang in equilibrium The coefficient of friction between the chain and table is _________ 

(A) [ℓL / (L + ℓ)] 

(B) (L/ℓ) 

(C) (ℓ/L) 

(D) [ℓ / (L – ℓ)]

Answer 1

Correct option: (D) [ℓ / (L – ℓ)]

Explanation:

Let mass be M.

Total length of chain = L

maximum possible length hanging = ℓ

mass per unit length = (M / L)

For portion of length (L – ℓ), frictional force = fs

fs = μ × (M / L) (L – ℓ) × g --- (fs = μmg)

For portion of length ℓ. force = mass × g

f = (M / L) × ℓ × g (f = mg)

both forces must be equal

μ (M / L) (L – ℓ)g = (M / L)ℓg

μ = [ℓ / (L – ℓ)]