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 0.130 rad/s2. After making 2844 revolutions, its angular speed is 136 rad/s. (a) What is the initial angular velocity of the turbine? (b) How much time elapses while the turbine is speeding up?

d = 2844rev * 6.28rad/rev = 17,860 radians.

a. V^2 = Vo^2 + 2a*d.
V = 136 rad/s, a = 0.130 rad/s, Vo = ?.

b. V = Vo + a*t. V = 136 rad/s, a = 0.130 rad/s^2, t = ?.

2

To solve this problem, we need to use the equations of angular motion. The first equation relates the final angular velocity (ωf), initial angular velocity (ωi), angular acceleration (α), and time (t):

ωf = ωi + αt (Equation 1)

The second equation relates the number of revolutions (N), angular displacement (θ), and number of radians per revolution (2π):

θ = 2πN (Equation 2)

Lastly, the third equation relates angular displacement (θ), initial angular velocity (ωi), final angular velocity (ωf), and angular acceleration (α):

θ = ωi*t + 0.5α*t^2 (Equation 3)

Now let's solve the problem step by step:

(a) What is the initial angular velocity of the turbine?

To find the initial angular velocity (ωi), we can rearrange Equation 3:

θ = ωi*t + 0.5α*t^2

Since the turbine initially spins at a constant angular speed, its initial angular displacement (θ) is 0. Using this information, the equation becomes:

0 = ωi*t + 0.5α*t^2

Since the initial angular displacement (θ) is 0, the equation simplifies to:

0 = ωi*t

This implies that either ωi = 0 or t = 0. However, since we are given that the turbine is initially spinning, the initial angular velocity (ωi) cannot be zero. Therefore, we conclude that t = 0.

Hence, the initial angular velocity (ωi) of the turbine is zero.

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

To find the time (t) the turbine takes to speed up, we can use Equation 1:

ωf = ωi + αt

Substituting the given values, we get:

136 rad/s = 0 + 0.130 rad/s^2 * t

Simplifying the equation, we can isolate t by dividing both sides by 0.130 rad/s^2:

t = 136 rad/s / 0.130 rad/s^2

t ≈ 1046.15 s

Therefore, the time elapsed while the turbine is speeding up is approximately 1046.15 seconds.