(a) The angular momentum (L) of a rotating object can be calculated using 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. We can do this using the formula:

m = W/g

where W is the weight and g is the acceleration due to gravity. In the English system, g 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, the moment of inertia is:

I = (4.658 slugs)(0.792 ft)^2 = 2.92 slug*ft^2

Next, we need to convert the rotational speed from rpm to rad/s. We can do this using the conversion factor 2π rad/rev and 1 min/60 s.

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

Finally, we can calculate the angular momentum:

L = (2.92 slug*ft^2)(183.26 rad/s) = 535.11 slug*ft^2/s

(b) The torque (τ) required to change the angular momentum of an object can be calculated using the formula:

τ = ΔL/Δt

where ΔL is the change in angular momentum and Δt is the change in time.

First, we need to calculate the final angular momentum at 800 rpm.

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

L_final = (2.92 slug*ft^2)(83.78 rad/s) = 244.63 slug*ft^2/s

The change in angular momentum is then:

ΔL = L_final - L_initial = 244.63 slug*ft^2/s - 535.11 slug*ft^2/s = -290.48 slug*ft^2/s

The change in time is 2 seconds, so the torque required is:

τ = ΔL/Δt = -290.48 slug*ft^2/s / 2 s = -145.24 lb*ft

The negative sign indicates that the torque is acting in the opposite direction of the initial rotation.

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