At t = 0 s a flywheel is rotating at 30 rpm. A motor gives it a constant acceleration of 0.9 rad/s^2 until it reaches 70 rpm. The motor is then disconnected. How many revolutions are completed at t = 20 s?
To find the number of revolutions completed at t = 20 s, we need to calculate the angular displacement of the flywheel during this time period.
We are given that initially, at t = 0 s, the flywheel is rotating at 30 rpm. Since 1 revolution is equal to 2π radians, we can convert 30 rpm to radians per second:
30 rpm = 30 * 2π/60 rad/s = π rad/s
Next, we know that the flywheel undergoes a constant acceleration of 0.9 rad/s^2 until it reaches 70 rpm. We can calculate the time it takes for the flywheel to reach 70 rpm using the formula for angular acceleration:
ω^2 = ω0^2 + 2αθ
Here, ω = final angular speed (70 rpm), ω0 = initial angular speed (30 rpm), α = angular acceleration (0.9 rad/s^2), and θ = angular displacement.
Converting the angular speeds to radians per second:
ω = 70 * 2π/60 rad/s = 7π/3 rad/s
ω0 = 30 * 2π/60 rad/s = π rad/s
Plugging these values into the formula, we have:
(7π/3)^2 = (π)^2 + 2 * 0.9 * θ
49π^2/9 = π^2 + 1.8θ
Simplifying, we get:
49π^2/9 - π^2 = 1.8θ
45π^2/9 = 1.8θ
5π^2 = θ
So, the angular displacement of the flywheel when it reaches 70 rpm is 5π^2 radians.
Finally, we need to calculate the angular displacement at t = 20 s. Since the motor is disconnected at t = 0 s, the flywheel will continue rotating at a constant angular velocity after that. The angular velocity is 7π/3 rad/s as calculated earlier.
Using the formula for angular displacement:
θ = ω0 * t + (1/2) * α * t^2
Here, ω0 = initial angular velocity (7π/3 rad/s), t = time (20 s), and α = 0 since there is no acceleration after the motor is disconnected.
Plugging in the values, we have:
θ = (7π/3) * 20 + (1/2) * 0 * (20)^2
θ = (7π/3) * 20
θ = 140π/3 rad
Therefore, the angular displacement of the flywheel at t = 20 s is 140π/3 radians.
To find the number of revolutions completed, we divide the angular displacement by 2π radians (1 revolution):
Number of revolutions = (140π/3)/(2π) = 70/3
So, the flywheel completes approximately 23.33 revolutions at t = 20 s.
1 rpm = 2pi rad/min = 2pi/60 rad/sec = pi/30 rad/sec = .1047 rad/sec
So, to increase from 30rpm to 70rpm is a change of 40rpm = 4.1888 rad/sec
This will take 4.1888/.9 seconds at .9rad/s^2 = 4.65 sec
During this 4.65 sec, it will complete 1/2 * .9 * 4.65^2 = 9.7 rad = 1.55 rev
The remaining 15.35 sec will complete 15.35*7.329 = 112.5 rad = 17.9 rev
So, the total revolutions = 19.5