Answer:
[tex]24.81\ \text{rad/s}^2[/tex]
Explanation:
M = Mass of cylinder = 4.31 kg
R = Radius of cylinder = 0.294 m
m = Mass of bucket = 6.27 kg
g = Acceleration due to gravity = [tex]9.81\ \text{m/s}^2[/tex]
[tex]\alpha[/tex] = Angular acceleration
a = Acceleration = [tex]\alpha R[/tex]
I = Moment of inertia of cylinder = [tex]\dfrac{MR^2}{2}[/tex]
The force balance of the system is
[tex]mg-T=ma\\\Rightarrow T=m(g-a)[/tex]
For the disk
[tex]TR=I\alpha\\\Rightarrow m(g-a)R=\dfrac{1}{2}MR^2\alpha\\\Rightarrow \alpha=\dfrac{g}{\dfrac{MR}{2m}+R}\\\Rightarrow \alpha=\dfrac{9.8}{\dfrac{4.31\times 0.294}{2\times 6.27}+0.294}\\\Rightarrow \alpha=24.81\ \text{rad/s}^2[/tex]
The angular acceleration of the pulley is [tex]24.81\ \text{rad/s}^2[/tex].
