Answer: 
Explanation:
Given
Initial angular speed is 
Final angular speed is 
Time period 
Magnitude of the fan's acceleration is given by

Insert the values

Thus, fan angular acceleration is 
Answer:
The speed is
.
(a) is correct option.
Explanation:
Given that,
Potential difference 
Speed 
If it were accelerated instead
Potential difference 
We need to calculate the speed
Using formula of initial work done on proton

We know that,


Put the value into the formula

....(I)
If it were accelerated instead through a potential difference of
, then it would gain a speed will be given as :
Using an above formula,

Put the value of 



Hence, The speed is
.
The maximum allowable torque must correspond to the allowable shear stress for maximization. To solve this, we use the torsion formula:
Max. Allowable Shear Stress = Maximum Torque ÷ Cross-Sectional Area
8 x 10^6 Pa = Maximum Torque ÷ pi*(d/2)²
Maximum Torque = 8 x 10^6 Pa * pi*(0.06/2)² m²
Maximum Torque = 22,619.47 J or
Maximum Torque = 22.62 kJ
As for the second question, I have no reference figure so I am unable to answer it. I hope I was still able to help you, though.