The difficulty is to find the acceleration with which the bucket will fall. It will not fall with "g" because of the tension in the rope.
On pole torque ,τ = Tr = Iα where I is moment of inertia of pole =Mr^2/2 and r is the radius of the pole
α = Tr/I ..........................................(1)
Applying newtons law on bucket,
ma = mg-T a=g-T/m .................................................(2)
If the rope does not slip
a=αr
g-T/m = Tr^2/I
Putting I = Mr^2/2 ,we get
g-T/m = 2T/M T = g/(1/m+2/M)
Putting this value of T in eqn (2), we will get the value of "a" the downward accelration of bucket.
(Please find a by putting the values of m and M yourself).
Now that we know the value of "a", find time taken to travel 20m is very easy.
Apply the equation of motion s=ut+1/2at^2
u=0
s=20m
I hope my answer has come to your help. Have a nice day ahead and may God bless you always!
