Section 8.3 The Equations of Rotational Kinematics

A wind turbine is initially spinning at a constant angular speed. As the wind's strength gradually increases, the turbine experiences a constant angular acceleration of 0.190 rad/s2. After making 2860 revolutions, its angular speed is 132 rad/s.

(a) What is the initial angular velocity of the turbine?
in rad/s

(b) How much time elapses while the turbine is speeding up?
in seconds

If you are familiar with the timeless equation in regular kinematics, you can also use it for this problem

(vf)² = (vi)² + 2ad
This transfers over to
(ωf)² = (ωi)² + 2αθ

ω is angular velocity
α is angular acceleration
θ is angle traveled in radians

a)
We know that the final angular velocity (ωf) is 132 rad/s.
It gives us the angular acceleration (α), which is 0.190 rad/s²
The angle traveled in radians can be found from 2860 revolutions. Just multiply 2,860 by 2π to convert revolutions to radians, so θ = 5720π rad

We have everything we need to solve for ωi, the initial angular velocity
(132 rad/s)² = (ωi)² + 2(0.190 rad/s²)(5720π rad)
ωi = 103 rad/s

b)
We know the angular acceleration and we know the initial and final angular velocities, so we can solve for t with the equation:
Δω = αt
(132 - 103 rad/s) = (0.190 rad/s²)t
t = 153 s