At t = 0 s a flywheel is rotating at 25 rpm. A motor gives it a constant acceleration of 0.8 rad/s2 until it reaches 100 rpm. The motor is then disconnected. How many revolutions are completed at t = 23 s?

The average speed during acceleration is 37.5 rpm.

The time spent accelerating is
(75 rpm)*(2 pi/60 rad/s*rpm)/0.8 rad/s^2= 9.8 seconds = 1.63 min

The number of revolutions turned while accelerating is
37.5*1.63 = 6.12

Add to that the number of revolutions turned after the motor is disconnected. They should tell you if it remains at constant speed after that. Friction will tend to slow it down, and they provide no information on that

Why did the flywheel go to therapy? It needed to unwind!

Now, let's calculate the number of revolutions completed at t = 23s.

To do that, we first need to find the time it takes for the flywheel to reach 100 rpm from 25 rpm, while accelerating at 0.8 rad/s^2.

We can use the equation:

ωf = ωi + αt

where ωf is the final angular velocity, ωi is the initial angular velocity, α is the angular acceleration, and t is the time.

Converting the given angular velocities into rad/s:

25 rpm = 25 * (2π/60) rad/s = 5π/6 rad/s
100 rpm = 100 * (2π/60) rad/s = 10π/3 rad/s

Plugging in the values:

10π/3 rad/s = 5π/6 rad/s + 0.8 rad/s^2 * t

Simplifying:

10π/3 rad/s - 5π/6 rad/s = 0.8 rad/s^2 * t
(20π - 5π)/6 rad/s = 0.8 rad/s^2 * t
15π/6 rad/s = 0.8 rad/s^2 * t
5π/2 rad/s = 0.8 rad/s^2 * t

Solving for t:

t = (5π/2 rad/s) / (0.8 rad/s^2)
t = (5π/2) / 0.8 s
t ≈ 6.25 s

So it takes approximately 6.25 seconds for the flywheel to reach 100 rpm.

Now, let's find the number of revolutions completed at t = 23s.

We'll use the equation:
θ = ωit + 0.5αt^2

where θ is the angular displacement, ωi is the initial angular velocity, α is the angular acceleration, and t is the time.

Plugging in the given values:

θ = (5π/6 rad/s) * (23 s) + 0.5 * (0.8 rad/s^2) * (23 s)^2

Simplifying:

θ = (5π/6 rad/s) * (23 s) + 0.5 * (0.8 rad/s^2) * (529 s^2)
θ = (115π/6 rad) + (0.5 * 0.8 * 529 π/6 rad)
θ = (115π/6 rad) + (0.4 * 529 π/6 rad)
θ = (115π + 211.6π) / 6 rad
θ = (326.6π/6) rad

Now, let's convert it to revolutions:

θ = (326.6π/6) rad * (1 rev / 2π rad)
θ = (326.6/6) rev
θ ≈ 54.43 rev

Therefore, at t = 23s, the flywheel completes approximately 54.43 revolutions. Hope that answer wasn't too round-about for you!

To find the number of revolutions completed at t = 23 s, we need to break down the problem into two parts:

1. The period during which the motor is connected.
2. The period after the motor is disconnected.

Let's calculate each of these periods step by step.

First, let's find the initial angular velocity (ω1) at t = 0 s. We know that the flywheel is rotating at 25 rpm, which is equivalent to 25 * 2π rad/min.

ω1 = 25 * 2π rad/min.

Next, let's find the final angular velocity (ω2) when the motor is disconnected. We know that the flywheel reaches 100 rpm, which is equivalent to 100 * 2π rad/min.

ω2 = 100 * 2π rad/min.

Now, let's compute the time it takes for the motor to accelerate the flywheel from ω1 to ω2. We can use the formula:

ω2 = ω1 + α * t,

where ω1 is the initial angular velocity, ω2 is the final angular velocity, α is the angular acceleration, and t is the time.

Plugging in the values:

100 * 2π = (25 * 2π) + (0.8 * t).

Simplifying the equation:

75 * 2π = 0.8 * t.

Divide both sides of the equation by 0.8:

t = (75 * 2π) / 0.8.

t = 589.0488 seconds (rounded to four decimal places).

Therefore, it takes approximately 589.0488 seconds for the motor to accelerate the flywheel.

Since we are interested in the number of revolutions at t = 23 s, we need to calculate the number of revolutions during the period when the motor is connected.

Let's call the number of revolutions during this period N1.

We can use the formula:

N = ω1 * t1 / (2π),

where N is the number of revolutions, ω1 is the initial angular velocity, t1 is the time, and 2π represents one full revolution.

Plugging in the values:

N1 = (25 * 2π) * 589.0488 / (2π).

Simplifying the equation:

N1 = 25 * 589.0488.

N1 = 14,726.22 (rounded to two decimal places).

Therefore, the number of revolutions completed during the period when the motor is connected is approximately 14,726.22 revolutions.

Next, let's calculate the number of revolutions after the motor is disconnected.

We have already found that it takes approximately 589.0488 seconds for the motor to accelerate the flywheel. Therefore, at t = 23 s, the motor has been disconnected for 23 - 589.0488 = -566.0488 seconds.

During this period, the angular velocity remains constant at ω2 = 100 * 2π rad/min.

Using the same formula as before:

N = ω2 * t2 / (2π),

where N is the number of revolutions, ω2 is the final angular velocity, and t2 is the time.

Plugging in the values:

N2 = (100 * 2π) * (-566.0488) / (2π).

Simplifying the equation:

N2 = -56,604.88 (rounded to two decimal places).

Therefore, the number of revolutions completed after the motor is disconnected is approximately -56,604.88 revolutions.

Finally, to find the total number of revolutions at t = 23 s, we add the number of revolutions during the period when the motor is connected (N1) and after the motor is disconnected (N2):

Total = N1 + N2.

Total = 14,726.22 + (-56,604.88).

Total = -41,878.66 (rounded to two decimal places).

Therefore, at t = 23 s, approximately -41,878.66 revolutions have been completed.

To find the number of revolutions completed at t = 23 s, we need to determine the final angular velocity and the initial angular velocity at t = 23 seconds.

Let's break down the problem step by step:

Step 1: Convert the given angular velocity from rpm to rad/s.
The initial angular velocity (ωi) at t = 0 s is given as 25 rpm (revolutions per minute). To convert it to rad/s, we need to multiply by the conversion factor: 1 rpm = (2π/60) rad/s. Therefore,
ωi = 25 rpm * (2π/60) rad/s = 25 * (2π/60) rad/s.

Step 2: Find the time it takes for the flywheel to reach 100 rpm.
The final angular velocity (ωf) is given as 100 rpm. We need to find the time it takes to reach this angular velocity by using the equation:
ωf = ωi + α * t,
where ωf is the final angular velocity, ωi is the initial angular velocity, α is the angular acceleration, and t is the time.

Since the motor supplies a constant acceleration of 0.8 rad/s^2, plugging in the values:
100 rpm = (25 * 2π/60) rad/s + (0.8 rad/s^2 * t1),
where t1 is the time taken to reach 100 rpm.

Solving for t1:
(0.8 rad/s^2 * t1) = 100 rpm - (25 * 2π/60) rad/s
(0.8 rad/s^2 * t1) ≈ 10.471 rad/s
t1 ≈ 10.471 rad/s / 0.8 rad/s^2

Step 3: Find the number of revolutions at t = 23 s.
Now, we have the initial angular velocity at t = 0 s and the time taken to reach 100 rpm. We can use the equation:
θ = ωi * t + (1/2) * α * t^2,
where θ is the angular displacement, ωi is the initial angular velocity, α is the angular acceleration, and t is the time.

Since the motor is disconnected after reaching 100 rpm, there is no angular acceleration (α) for t > t1.

Using this equation:
θ = (25 * 2π/60) rad/s * 23 s + (1/2) * 0.8 rad/s^2 * (23 s)^2

Simplifying the equation:
θ = (25 * 2π/60 * 23) + (1/2) * 0.8 * (23)^2

Finally, calculating the value of θ will give us the number of revolutions completed at t = 23 s.