Question

A pulley of the radius R and mass M has an Inextensible string placed over it. two masses M₁ and M₂ (M₂ > M₁) are attached to the free ends of the string. Find 

1. The acceleration of masses the system 

2. Tension in the string.

Answer 1

Consider the free body diagram (FBD)

T₁ and T₂ are the tension at the two parts of the string N is normal force provided at the hinge. considering equation ΣF_ext = ma

m₂g – T₂ = m₂a ……….. (1)

m₁g – T₁ = m₁a ………. (2)

[Since the 2 masses are connected by string they have same displacement hence same acceleration]

R α_pulley = a – (3)

(condition for rolling of pulley)

(T₂ – T₁) R = l α_pulley = \(\frac{MR^2}{2}\) α_pulley ………. (4)

from (3) and (4)

T₂ – T₁ = \(\frac {Ma}{2}\) ……….. (5)

from (1) and (2)

T₂ = m₂ (g – a)

and T₁ = m₁ (g + a)

∴ T₂ – T₁ = m₂ (g – a) – m₁ (g + a) = \(\frac {Ma}{2}\)

⇒ g(m₂ – m₁) = a [ \(\frac {M}{2}\)+ m₁ + m₂]