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 0.180 rad/s2. After making 2874 revolutions, its angular speed is 139 rad/s.
A. What is the initial angular velocity of the turbine?
B. How much time elapses while the turbine is speeding up?
wf^2=wi^2+2ad solve for wi, d= 2874*2PI, a is given, wf is given
wf=wi+at solve for t
To solve this problem, we can use the equations of angular motion. The first equation relates angular displacement (θ), initial angular velocity (ω0), angular acceleration (α), and time (t):
θ = ω0 * t + 0.5 * α * t^2
The second equation relates final angular velocity (ω), initial angular velocity (ω0), angular acceleration (α), and angular displacement (θ):
ω^2 = ω0^2 + 2 * α * θ
We are given the following information:
α = 0.180 rad/s^2
θ = 2874 revolutions = 2874 * 2π radians (since 1 revolution = 2π radians)
ω = 139 rad/s
A. To find the initial angular velocity (ω0), we can use the second equation. Rearranging it, we get:
ω0^2 = ω^2 - 2 * α * θ
ω0^2 = (139 rad/s)^2 - 2 * (0.180 rad/s^2) * (2874 * 2π rad)
ω0^2 = 19321 - 3275.2π
ω0 ≈ √(19321 - 3275.2π) rad/s
Thus, the initial angular velocity of the turbine is approximately √(19321 - 3275.2π) rad/s.
B. To find the time elapsed while the turbine is speeding up, we can use the first equation. Rearranging it, we get:
θ - 0.5 * α * t^2 = ω0 * t
(2874 * 2π rad) - 0.5 * (0.180 rad/s^2) * t^2 = √(19321 - 3275.2π) rad/s * t
Simplifying this equation and solving for t involves numerical methods or approximations.