1)

2) 3.0 s
Step-by-step explanation:
1)
The angular acceleration of a rigid body in rotation can be found by using the equivalent of Newton's second law for rotational motions:
(1)
where
is the torque on the object
I is the moment of inertia of the object
is the angular acceleration
Here we have:
is the moment of inertia of a solid sphere about its central axis, where
M = 240 g = 0.240 kg is the mass of the sphere
R = 4.50 cm /2= 2.25 cm = 0.0225 m is the radius of the sphere
is the torque exerted by the frictional force, where
is the force of friction (negative because the direction is opposite to the motion)
r = R = 0.0225 m is the distance of the point of application of the force from the centre
Substituting into eq(1) we find

And solving for
, we find the angular acceleration:

2)
Here we know that the motion of the sphere is an angular accelerated motion.
Therefore, we can use the equivalent of suvat equations for rotational motion:

where
is the final angular velocity
is the initial angular velocity
is the angular acceleration
t is the time
In this problem, we know that
, since we are told that the rotational speed decreases by 28.0 rad/s
is the angular acceleration of the sphere
Solving for t, we find how long it takes for the sphere to decelerate by 28.0 rad/s:
