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

Step-by-step 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 (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=(\omega-\omega_0)/(\alpha)=(140-126.59)/(0.1)`

`t=134.1 s`