A Micro –Hydro turbine generator rotor is accelerating uniformly from an angular velocity of 610 rpm to its operating angular velocity of 837 rpm. The radius of the rotor is 0.62 m and its rotational acceleration is 5.9 rad/s2. What is the rotor’s angular velocity (in rad/s) after 3.2 s?
To find the rotor's angular velocity after 3.2 seconds, we need to know its initial angular velocity, final angular velocity, and the time taken to accelerate.
Given:
Initial angular velocity, ω1 = 610 rpm
Final angular velocity, ω2 = 837 rpm
Rotational acceleration, α = 5.9 rad/s^2
Time, t = 3.2 s
We need to convert the angular velocities from rpm to rad/s:
ω1 = (610 rpm) * (2π rad/1 min) * (1 min/60 s) = 64.025 rad/s
ω2 = (837 rpm) * (2π rad/1 min) * (1 min/60 s) = 87.686 rad/s
Using the rotational kinematic equation:
ω2 = ω1 + α * t
Substituting the known values:
87.686 rad/s = 64.025 rad/s + (5.9 rad/s^2) * (3.2 s)
87.686 rad/s = 64.025 rad/s + 18.88 rad/s
Simplifying the equation:
87.686 rad/s - 64.025 rad/s = 18.88 rad/s
23.661 rad/s ≈ 18.88 rad/s
Therefore, the rotor's angular velocity after 3.2 seconds is approximately 23.661 rad/s.
To find the rotor's angular velocity after 3.2 seconds, we need to use the formula for angular acceleration:
Angular acceleration = (Angular velocity final - Angular velocity initial) / time
Given:
Angular velocity initial = 610 rpm
Angular velocity final = 837 rpm
Time = 3.2 seconds
First, we need to convert the angular velocities from rpm to rad/s. Since 1 rpm = (2π/60) rad/s, we can convert as follows:
Angular velocity initial = 610 rpm * (2π/60) rad/s = 64.03 rad/s
Angular velocity final = 837 rpm * (2π/60) rad/s = 87.59 rad/s
Now we can substitute the values into the formula:
Angular acceleration = (87.59 rad/s - 64.03 rad/s) / 3.2 s
Angular acceleration = 23.56 rad/s / 3.2 s
Angular acceleration = 7.3625 rad/s²
Now, we know the angular acceleration and the time, so we can find the angular velocity after 3.2 seconds by using the formula:
Angular velocity final = Angular velocity initial + (Angular acceleration * time)
Angular velocity final = 64.03 rad/s + (7.3625 rad/s² * 3.2 s)
Angular velocity final = 64.03 rad/s + 23.56 rad/s
Angular velocity final = 87.59 rad/s
Therefore, the rotor's angular velocity after 3.2 seconds is 87.59 rad/s.
V1 = 610rev/min * 6.28rad/rev * 1min/60s = 63.85 rad/s.
V2 = V1 + a*t. = 63.85 + 5.9*3.2 = 82.73 m/s.