At t = 0 s a flywheel is rotating at 20 rpm. A motor gives it a constant acceleration of 1 rad/s^2 until it reaches 100 rpm. The motor is then disconnected. How many revolutions are completed at t = 21 s?
What I've done so far is multiply my 20rpm and 100rpm by (2pi/60) to get what I think is my Wo and Wf. I tried using kinematics with wf=wo+ang.acc x t and plugged in those values to find a t of 8.38 s. So I'm thinking it takes 8.38 s to get to 100rpm. I tried doing a proportion with 20rpm and 8.38s , and then one with 100rpm and (21-8.38s) , to then add the total rotations but that's not right.
To solve this problem, let's break it down step by step:
Step 1: Find the time it takes for the flywheel to reach 100 rpm.
We can use the equation of motion for rotational kinematics:
ω_f = ω_0 + αt,
where ω_f is the final angular velocity (100 rpm), ω_0 is the initial angular velocity (20 rpm), α is the constant angular acceleration (1 rad/s^2), and t is the time it takes to reach ω_f.
First, convert both angular velocities from rpm to rad/s:
ω_f = 100 rpm * 2π/60 = 10π/3 rad/s,
ω_0 = 20 rpm * 2π/60 = 2π/3 rad/s.
Now, plug these values into the equation:
10π/3 = 2π/3 + 1 * t.
Simplifying, we get:
8π/3 = t.
So it takes 8π/3 seconds for the flywheel to reach 100 rpm.
Step 2: Calculate the number of revolutions completed at t = 21 s.
To find the number of revolutions, we need to know the angular displacement.
Since we have constant angular acceleration, we can use the equation:
θ = ω_0 * t + (1/2) * α * t^2,
where θ is the angular displacement, ω_0 is the initial angular velocity, α is the angular acceleration, and t is the time.
Substituting the known values:
θ = (2π/3) * 8π/3 + (1/2) * 1 * (8π/3)^2.
Simplifying:
θ = 16π^2 / 9 + 64π^2 / 9.
θ = 80π^2 / 9.
Now, we can find the number of revolutions by dividing the angular displacement by 2π:
revolutions = (80π^2 / 9) / 2π = 40π / 9.
So, at t = 21 s, the flywheel completes 40π / 9 revolutions.
To solve this problem, let's break it down into two parts.
Part 1: Acceleration phase
From your calculations, you have determined that it takes 8.38 seconds to go from 20 rpm to 100 rpm with a constant acceleration of 1 rad/s^2.
Using the formula for angular velocity with constant acceleration:
ωf = ωi + αt
where:
ωf = final angular velocity (100 rpm in this case)
ωi = initial angular velocity (20 rpm in this case)
α = angular acceleration (1 rad/s^2 in this case)
t = time taken (unknown)
Converting angular velocities to rad/s:
ωf = 100 rpm * (2π rad/1 min) * (1 min/60 s) = 10π rad/s
ωi = 20 rpm * (2π rad/1 min) * (1 min/60 s) = 2π/3 rad/s
Now we can plug in these values and solve for t:
10π = 2π/3 + 1 * t
Multiplying both sides by 3:
30π = 2π + 3t
28π = 3t
t = 28π/3 ≈ 29.32 s
So it takes approximately 29.32 seconds to go from 20 rpm to 100 rpm.
Part 2: Deceleration phase
We know that at t = 8.38 seconds, the motor is disconnected and the flywheel continues rotating at a constant speed of 100 rpm.
We now need to determine the number of revolutions completed during the time interval from t = 8.38 s to t = 21 s.
To do this, we need to calculate the angular displacement during this time interval.
Using the formula for angular displacement:
θ = ωi * t + (1/2) * α * t^2
where:
θ = angular displacement (unknown)
ωi = initial angular velocity (100 rpm in this case)
t = time taken (21 s - 8.38 s = 12.62 s in this case)
α = angular acceleration (0 rad/s^2 since the motor is disconnected)
Converting angular velocity to rad/s:
ωi = 100 rpm * (2π rad/1 min) * (1 min/60 s) = 10π/3 rad/s
Plugging in these values, we get:
θ = (10π/3) * 12.62 + (1/2) * 0 * (12.62)^2
θ ≈ 133π rad
To convert this angular displacement to revolutions, we divide by 2π:
Number of revolutions = (133π rad) / (2π rad/rev) ≈ 133/2 ≈ 66.5 revolutions
Therefore, at t = 21 s, the flywheel completes approximately 66.5 revolutions.