Final answer:
The value of the angular acceleration α is approximately -0.314 rad/s² and the initial angular velocity ωi is 78.0 rad/s.
Step-by-step explanation:
To find the angular acceleration (α) and the initial angular velocity (ωi), we can use the following equations:
ωf = ωi + αt
θ = ωi*t + (1/2)*α*t^2
Given that ωf = 75.9 rad/s, t = 10 s, and θ = 37.0 revolutions = 2π*37.0 rad, we can substitute these values into the equations:
75.9 = ωi + α*10
2π*37.0 = ωi*10 + (1/2)*α*10^2
Simplifying these equations, we get:
ωi + 10α = 75.9
37π = 10ωi + 50α
Now, we have a system of equations that we can solve for ωi and α. Subtracting the first equation from the second equation, we get:
37π - ωi = 10ωi + 50α - 75.9
Combining like terms, we get:
11ωi + 50α = 37π - 75.9
Substituting the value of ωi from the first equation into the second equation, we get:
11(75.9 - 10α) + 50α = 37π - 75.9
Simplifying and rearranging the equation, we obtain:
527.9 - 110α + 50α = 37π - 75.9
Combining like terms, we get:
527.9 - 60α = 37π - 75.9
Subtracting 527.9 from both sides and rearranging the equation, we have:
-60α = 37π - 603.8
Dividing both sides by -60, we get:
α ≈ -0.314 rad/s²
Substituting this value of α into the first equation, we can solve for ωi:
75.9 = ωi + (-0.314)*10
Simplifying the equation, we get:
ωi = 78.0 rad/s
Therefore, the value of the angular acceleration α is approximately -0.314 rad/s² and the initial angular velocity ωi is 78.0 rad/s.