Question

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 0.100 rad/s2. After making 2844 revolutions, its angular speed is 140 rad/s.

Answer 1

Complete answer

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 0.100rad/s2. After making 2844 revolutions, its angular speed is 140rad/s

(a) What is the initial angular velocity of the turbine? (b) How much time elapses while the turbine is speeding up?

Answer:

a) 126.59 radians per second

b) 134.1 seconds

Explanation:

We can use the rotational kinematic equations for constant angular acceleration.

a) For a) let’s use:

$\omega^{2}=\omega_{0}^{2}+2\alpha\varDelta\theta$ (1)

with $\omega_{0}$ the initial angular velocity, $\omega$ the final angular velocity, $\alpha$ the angular acceleration and $\Delta \theta$the revolutions on radians (2844 revolutions = 17869.38 radians). Solving (1) for initial velocity:

$\sqrt{\omega^{2}-2\alpha\varDelta\theta}=\omega_{0}$

$\omega_{0}^2=\sqrt{(140)^2 -(2)(0.100)(17869.38)=126.59 \frac{rad}{s}}$

b) Knowing those values, we can use now the kinematic equation

$\omega=\omega_{0}+\alpha t$

with t the time, solving for t:

$t=\frac{\omega-\omega_0}{\alpha}=\frac{140-126.59}{0.1}$

$t=134.1 s$