To solve this problem we will apply the concepts related to the balance of Forces, the centripetal Force and Newton's second law.
I will also attach a free body diagram that allows a better understanding of the problem.
For there to be a balance between weight and normal strength, these two must be equivalent to the centripetal Force, therefore
[tex]F_c = W-N[/tex]
[tex]m\omega^2r = W-N[/tex]
Here,
m = Net mass
[tex]\omega[/tex]= Angular velocity
r = Radius
W = Weight
N = Normal Force
[tex]m\omega^2r = 174-146[/tex]
The net mass is equivalent to
[tex]F = mg \rightarrow m = \frac{F}{g}[/tex]
Then,
[tex]m = \frac{174lb}{32.17ft/s^2}[/tex]
Replacing we have then,
[tex](\frac{174lb}{32.17ft/s^2})\omega^2 (54ft) =174lb-146lb[/tex]
Solving to find the angular velocity we have,
[tex]\omega = 0.309rad/s[/tex]
Therefore the angular velocity is 0.309rad/s
