Falling solid cylinder with string

[haruspex]

Torque= T R
=m(g-a)R

This is the first equation in your post #15:

T= 1/(2) (mr^2) a/r

As I wrote, the left hand side is wrong because it is a force, not a torque. Replace that T by the torque it exerts (as in post #33).

 

[Suyash Singh]

Torque=F R
Torue=I alpha
=1/2 MR^2 a/R

1/2 MR^2 a/R = F R
1/2 Ma=F
a=2F/m
a=2(g-a)
3a=2g

a=2g/3

Tension=m(g-a)
=m(g-2g/3)
T=mg/3

 

[haruspex]

Torque=F R
Torue=I alpha
=1/2 MR^2 a/R

1/2 MR^2 a/R = F R
1/2 Ma=F
a=2F/m
a=2(g-a)
3a=2g

a=2g/3

Tension=m(g-a)
=m(g-2g/3)
T=mg/3

That's it.