Answer:
Look below
Step-by-step explanation:
Part A:
The initial angular velocity of the rotor can be found using the formula:
ω = v/r
where ω is the angular velocity, v is the linear velocity, and r is the radius of the cylinder. Since the rotor is rotating at 9000 rpm, we can convert this to radians per second:
ωi = (9000 rpm) x (2π/60 s) = 942.48 rad/s
The final angular velocity is zero. The frictional torque acting on the rotor is given by:
τ = Iα
where τ is the torque, I is the moment of inertia, and α is the angular acceleration. The moment of inertia for a solid cylinder is:
I = (1/2)mr^2
Substituting in the given values and solving for α, we get:
α = τ/I = (1.73 m⋅N) / [(1/2)(3.72 kg)(0.0760 m)^2] = 371.7 rad/s^2
The final angular velocity is zero, so we can use the formula:
ωf^2 = ωi^2 + 2αθ
where θ is the angle of rotation. Solving for θ, we get:
θ = (ωf^2 - ωi^2) / (2α) = (0 - (942.48 rad/s)^2) / (2 x 371.7 rad/s^2) = 6.76 revolutions
Therefore, the rotor will turn approximately 6.76 revolutions before coming to rest.
Part B:
The time it takes for the rotor to come to rest can be found using the formula:
ωf = ωi + αt
where ωf is the final angular velocity, ωi is the initial angular velocity, and α is the angular acceleration. Solving for t, we get:
t = (ωf - ωi) / α = (0 - 942.48 rad/s) / 371.7 rad/s^2 = 2.535 s
Therefore, it will take approximately 2.535 seconds for the rotor to come to rest.