A uniform rope of mass m hangs freely from a ceiling. A bird of mass M climbs up the rope with an acceleration a. The force exerted by the rope on the ceiling is

Net force exerted by the rope on the ceiling = force exerted by the rope itself on the ceiling + force exerted by the bird on the rope

T = T₁​ + T₂​

We have,

T₁ ​= mg

T₂ − Mg = Ma

⇒T₂ = M(g + a)

∴T = M(g + a) + mg