This question involves the concepts of torque, angular acceleration, and the moment of inertia.
(a) The angular acceleration is "- 14.81 rad/s²".
(b) It will take "1.52 s" to decrease its rotational speed by 22.5 rad/s.
(a)
First, we will find out the moment of inertia of the sphere, by using the following formula:
[tex]I = \frac{2}{5}mr^2\\\\[/tex]
where,
m = mass = 225 g = 0.225 kg
r = radius = diameter/2 = 3 cm/2 = 1.5 cm = 0.015 m
Therefore,
[tex]I = \frac{2}{5}(0.225\ kg)(0.015\ m)^2\\\\[/tex]
I = 2.025 x 10⁻⁵ kg.m²
Now, we will be calculating the torque applied on the machine part:
[tex]T = -Fr[/tex] (negative sign due to resistive frictional force)
where,
T = torque = ?
F = frictional force = 0.02 N
Therefore,
[tex]T = -(0.02\ N)(0.015\ m)\\[/tex]
T = - 3 x 10⁻⁴ N.m
Now, we will calculate the angular acceleration of the machine part:
[tex]T = I\alpha\\\\\alpha=\frac{T}{I}\\\\\alpha=\frac{-3\ x\ 10^{-4}\ N.m}{2.025\ x\ 10^{-5}\ kg.m^2}\\\\\alpha=-14.81\ rad/s^2[/tex](negative sign shows deceleration)
(b)
Now, we will calculate the time taken to decrease the speed by using the simple formula of angular acceleration:
[tex]\alpha = \frac{\Delta \omega}{t}\\\\t= \frac{\Delta \omega}{\alpha}[/tex]
where,
Δω = change in angular speed = - 22.5 rad/s
t = time taken = ?
Therefore,
[tex]t=\frac{-22.5\ rad/s}{-14.81\ rad/s^2}[/tex]
t = 1.52 s
Learn more about torque here:
https://brainly.com/question/6855614?referrer=searchResults
The attached picture shows torque.
