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?
number of revolutions?
Well, it sounds like the flywheel went from zero to hero! Let's calculate the number of revolutions it completes at t = 20 s.
First, we need to convert the angular velocities from rpm to rad/s.
30 rpm = 30 * 2π/60 rad/s = π rad/s
70 rpm = 70 * 2π/60 rad/s = (7π/3) rad/s
Next, we can find the time it takes for the flywheel to reach 70 rpm using the equation:
ω = ω0 + αt
where ω0 is the initial angular velocity, α is the acceleration, and t is the time.
(7π/3) rad/s = π rad/s + (0.9 rad/s^2)t
Solving this equation, we find that t = 10 s.
Now, let's find the total number of revolutions completed at t = 20 s.
Since the motor is disconnected at t = 10 s, the flywheel retains its angular velocity of (7π/3) rad/s for the next 10 seconds.
Therefore, the total number of revolutions completed at t = 20 s is:
(7π/3) rad/s * 10 s / 2π rad = (7π/6) rev
And there you go! At t = 20 s, the flywheel completes (7π/6) revolutions.
To determine the number of revolutions completed at t = 20 s, we need to calculate the angular displacement of the flywheel over the given time period.
Given:
Initial angular velocity of the flywheel, ω1 = 30 rpm
Final angular velocity of the flywheel, ω2 = 70 rpm
Time, t = 20 s
Acceleration, α = 0.9 rad/s^2
First, we need to convert the initial and final angular velocities from rpm to rad/s. We know that 1 revolution = 2π radians.
ω1 = (30 rpm) * (2π rad/1 min) * (1 min/60 s)
ω1 = π rad/s
ω2 = (70 rpm) * (2π rad/1 min) * (1 min/60 s)
ω2 = (7π/3) rad/s
Now, we can find the angular displacement using the formula:
θ = ω1 * t + (1/2) * α * t^2
θ = π * 20 + (1/2) * 0.9 * (20)^2
θ = 20π + 9 * 20
θ = 20π + 180
θ = 20(π + 9) rad
Finally, we can convert the angular displacement to the number of revolutions:
Number of revolutions = θ / (2π)
Number of revolutions = 20(π + 9) / (2π)
Number of revolutions = 10(π + 9) / π
Therefore, at t = 20 s, the flywheel completes approximately 10(π + 9) / π revolutions.