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 0.165 rad/s2. After making 2870 revolutions, its angular speed is 143 rad/s. (a) What is the initial angular velocity of the turbine

The initial angular velocity is w0 = 120.42 rad/s

Explanation:

The initial speed is w(t) =w0

now, we have an acceleration of a = 0.165 rad/s^2, then after this point we have:

w(t) = 0.165 rad/s^2*t + w0

and for a given t = t0, we have that:

the relative position:

p(t) = (1/2)*(0.165rad/s^2)*t^2 + w0*t

each time that p(t) = 2*n*3.14, this means that the turbine did one revolution. (because in one revolution we have 2*3.14 rads)

then for t0 we have:

((1/2)*(0.165rad/s^2)*t0^2 + w0*t0)/(3.14*2) = 2870

this means that t0 is the time that the turbine needs to do 2870 revolutions.

and we also have that for t0:

w(t0) = 143 rad/s =  0.165 rad/s^2*t0 + w0

then we have two variables and two equations, we must solve the system

in the angular velocity eqatuion we can isolate w0 and get:

w0 = 143rad/s - 0.165 rad/s^2*t0

now we can replace t)his in the other equation and get:

(1/2)*(0.165rad/s^2)*t0^2 + (143rad/s - 0.165 rad/s^2*t0)*t0 = 2870*(3.14*2)

and we can solve this for t0.

(-0.0825rad/s^2)*t0^2 + 143rad/s*t0 - 18023.6 = 0

here we hace that:

t0 = (-143 +- √(143^2 - 4*0.0825*18023.6))/(-2*0.0825) seconds

t0 = ( -143 + - 120.42)/(-0.165) seconds

here we have two solutions for t0, one for each sign of the 120.42. Now, we need to take the smallest positive value of t0, this will be:

t0 = (-143 + 120.42)/(-0.165) seconds = 136.85 s

now with this time we can find the value of w0

w0 = 143rad/s - 0.165 rad/s^2*t0

w0 = 143rad/s - 0.165 rad/s^2*136.85s = 120.42 rad/s