A 4.50-kg wheel that is 34.5 cm in diameter rotates through an angle of 13.8 rad as it slows down

uniformly from 22.0 rad/s to 13.5 rad/s. What is the magnitude of the angular acceleration of
the wheel?
A) 0.616 rad/s
2
B) 22.5 rad/s
2
C) 5.45 rad/s
2
D) 10.9 rad/s
2
E) 111 rad/s
2

W2^2 = W1^2 + 2*alpha*theta

13.5^2 = 22.0^2 +2*alpha*13.8
alpha = (13.5^2 - 22.0^2)/2*13.8
= - 10.9 rad/s^2
So, magnitude of angular acceleration = 10.9 rad/s^2

Well, well, well, looks like we've got ourselves a wheel in motion conundrum! So, to find the magnitude of the angular acceleration, we need to gather some information. First off, we have the initial angular velocity of 22.0 rad/s and the final angular velocity of 13.5 rad/s.

Next up, we have the angle of rotation, which is 13.8 rad. This doesn't look like just any ol' angle, it's a radian-angle! Now, to calculate the magnitude of the angular acceleration, we can use the formula:

angular acceleration = (final angular velocity - initial angular velocity) / angle of rotation

Plugging in the values, we get:

angular acceleration = (13.5 rad/s - 22.0 rad/s) / 13.8 rad
≈ -8.5 rad/s / 13.8 rad
≈ -0.615 rad/s²

Now, we've got to be careful, since we're looking for the magnitude of the angular acceleration. The magnitude doesn't care about direction, so we just need to take the absolute value of our result:

magnitude of angular acceleration ≈ | -0.615 rad/s² |
≈ 0.615 rad/s²

So, after all that mathematical fun, the magnitude of the angular acceleration of the wheel is approximately 0.615 rad/s². And there you have it!

To find the angular acceleration, we need to use the formula:

ω^2 = ωi^2 + 2αθ

where ω is the final angular velocity, ωi is the initial angular velocity, α is the angular acceleration, and θ is the angle through which the wheel rotates.

We have:
ω = 13.5 rad/s
ωi = 22.0 rad/s
θ = 13.8 rad

Plugging the values into the formula, we get:

(13.5 rad/s)^2 = (22.0 rad/s)^2 + 2α(13.8 rad)

182.25 rad^2/s^2 = 484.0 rad^2/s^2 + 27.6 α rad

Rearranging the equation, we get:

27.6 α rad = 182.25 rad^2/s^2 - 484.0 rad^2/s^2

27.6 α rad = -301.75 rad^2/s^2

α = (-301.75 rad^2/s^2) / 27.6 rad

α ≈ -10.909 rad/s^2

Since we are looking for the magnitude of the angular acceleration, the answer is:

|α| ≈ 10.909 rad/s^2

Therefore, the correct answer is D) 10.9 rad/s^2.

To find the angular acceleration of the wheel, we can use the formula:

angular acceleration (α) = change in angular velocity (Δω) / change in time (Δt)

First, let's find the change in angular velocity (Δω). We know that the initial angular velocity (ω1) is 22.0 rad/s, and the final angular velocity (ω2) is 13.5 rad/s. Therefore:

Δω = ω2 - ω1
Δω = 13.5 rad/s - 22.0 rad/s
Δω = -8.5 rad/s

Next, we need to find the change in time (Δt) during which the wheel slows down. Since we don't have the time value, we need to calculate it using the given information.

To calculate the time, we can use the equation:

Angle (θ) = ω1 * Δt + 0.5 * α * Δt^2

The angle (θ) is given as 13.8 radians, and the initial angular velocity (ω1) is 22.0 rad/s. Also, we know that the wheel is slowing down uniformly, so we can assume a constant angular acceleration (α). We need to solve this equation for Δt.

Rearranging the equation and substituting the given values, we get:

0.5 * α * Δt^2 + ω1 * Δt - θ = 0

0.5 * α * Δt^2 + 22.0 rad/s * Δt - 13.8 rad = 0

This is a quadratic equation, and we can solve it by using the quadratic formula:

Δt = (-b ± √(b^2 - 4ac)) / (2a)

Here, a = 0.5, b = 22.0 rad/s, and c = -13.8 rad.

Solving for Δt, we find two possible solutions:

1. Δt = (-22.0 rad/s + √(22.0 rad/s)^2 - 4(0.5)(-13.8 rad)) / (2(0.5))
2. Δt = (-22.0 rad/s - √(22.0 rad/s)^2 - 4(0.5)(-13.8 rad)) / (2(0.5))

Calculating these values, we find:

1. Δt ≈ 1.61 seconds
2. Δt ≈ -3.23 seconds

Since time cannot be negative in this case, we take the positive solution for Δt. Therefore, Δt ≈ 1.61 seconds.

Finally, we can substitute the values of Δω and Δt into the formula for angular acceleration:

angular acceleration (α) = Δω / Δt
α = -8.5 rad/s / 1.61 s
α ≈ -5.28 rad/s²

The magnitude of the angular acceleration is the absolute value of α:

Magnitude of angular acceleration = |-5.28 rad/s²|
Magnitude of angular acceleration ≈ 5.28 rad/s²

Therefore, the correct answer is C) 5.45 rad/s².