(a) The angular momentum (L) of a rotating object is given by the formula L = Iω, where I is the moment of inertia and ω is the angular velocity.

The moment of inertia (I) for a rotating object is given by the formula I = mk^2, where m is the mass and k is the radius of gyration.

First, we need to convert the weight of the rotor to mass. Since weight = mass * gravity, we can find the mass by dividing the weight by the acceleration due to gravity. In the English system, gravity is approximately 32.2 ft/s^2. We also need to convert the weight from pounds to slugs (the unit of mass in the English system), and the radius of gyration from inches to feet.

m = 150 lb / 32.2 ft/s^2 = 4.658 slugs

k = 9.5 in * (1 ft / 12 in) = 0.792 ft

So, I = 4.658 slugs * (0.792 ft)^2 = 2.922 slug*ft^2

Next, we need to convert the rotational speed from rpm to rad/s.

ω = 1750 rpm * (2π rad / 1 rev) * (1 min / 60 s) = 183.26 rad/s

Finally, we can find the angular momentum:

L = Iω = 2.922 slug*ft^2 * 183.26 rad/s = 535.5 slug*ft^2/s

(b) The torque (τ) required to change the angular momentum of an object is given by the formula τ = ΔL / Δt, where ΔL is the change in angular momentum and Δt is the time over which the change occurs.

First, we need to find the final angular velocity when the rotor slows to 800 rpm.

ω_final = 800 rpm * (2π rad / 1 rev) * (1 min / 60 s) = 83.78 rad/s

Then, we can find the final angular momentum:

L_final = Iω_final = 2.922 slug*ft^2 * 83.78 rad/s = 244.8 slug*ft^2/s

The change in angular momentum is then ΔL = L_final - L = 244.8 slug*ft^2/s - 535.5 slug*ft^2/s = -290.7 slug*ft^2/s

Finally, we can find the torque:

τ = ΔL / Δt = -290.7 slug*ft^2/s / 2 s = -145.35 lb*ft

The negative sign indicates that the torque is acting in the opposite direction of the initial rotation, as expected for a slowing rotor.

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