A rotating wheel requires 8.00 s to rotate 29.0 revolutions. Its angular velocity at the end of the 8.00 s interval is 95.0 rad/s. What is the constant angular acceleration of the wheel? (Do not assume that the wheel starts at rest.)

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

Angular acceleration (α) = (Final angular velocity - Initial angular velocity) / Time

Given:
Final angular velocity (ωf) = 95.0 rad/s
Time (t) = 8.00 s

Step 1: Find the initial angular velocity (ωi)

Since we are not given the initial angular velocity, we need to calculate it using the formula:

Initial angular velocity (ωi) = (Number of revolutions / Time) × (2π radians / 1 revolution)

Given:
Number of revolutions = 29.0
Time (t) = 8.00 s

We convert the number of revolutions to radians by multiplying by 2π:

Initial angular velocity (ωi) = (29.0 revolutions / 8.00 s) × (2π radians / 1 revolution)

Step 2: Calculate the angular acceleration (α)

Now that we have the initial and final angular velocities, we can calculate the angular acceleration using the formula:

Angular acceleration (α) = (ωf - ωi) / t

Substituting the values:

Angular acceleration (α) = (95.0 rad/s - ωi) / 8.00 s

Step 3: Substitute the value of ωi obtained in Step 1

Substitute the value of initial angular velocity (ωi) obtained in Step 1 into the formula:

Angular acceleration (α) = (95.0 rad/s - ωi) / 8.00 s

Step 4: Calculate the angular acceleration (α)

Using the provided values, we can now calculate the angular acceleration:

Angular acceleration (α) = (95.0 rad/s - ωi) / 8.00 s

Substituting the obtained value of initial angular velocity (ωi):

Angular acceleration (α) = (95.0 rad/s - ωi) / 8.00 s

Hence, the constant angular acceleration of the wheel can be found using the equation (95.0 rad/s - ωi) / 8.00 s.

To find the constant angular acceleration of the wheel, we can use the following equation:

ω = ω₀ + αt

where:
ω is the final angular velocity (95.0 rad/s)
ω₀ is the initial angular velocity (unknown)
α is the angular acceleration (unknown)
t is the time interval (8.00 s)

The angular velocity can be related to the number of revolutions as follows:

ω = 2πn/t

where n is the number of revolutions (29.0), and t is the time interval (8.00 s).

Now, we can rearrange the equation ω = 2πn/t to find ω₀, the initial angular velocity:

ω₀ = ω - 2πn/t

Substituting the given values:

ω₀ = 95.0 rad/s - (2π)(29.0 revolutions) / 8.00 s

First, we need to convert revolutions to radians:

1 revolution = 2π radians

ω₀ = 95.0 rad/s - (2π)(29.0)(2π) / 8.00 s
ω₀ = 95.0 rad/s - (4π²)(29.0) / 8.00 s

Now we have the initial angular velocity ω₀. By substituting the values of ω, α, and t into the equation ω = ω₀ + αt, we can solve for α:

95.0 rad/s = ω₀ + α(8.00 s)

Substituting ω₀ into the equation:

95.0 rad/s = [95.0 rad/s - (4π²)(29.0) / 8.00 s] + α(8.00 s)

Now, we can solve for α by rearranging the equation:

α = (95.0 rad/s - [95.0 rad/s - (4π²)(29.0) / 8.00 s]) / 8.00 s

Simplifying further:

α = [(4π²)(29.0) / 8.00 s] / 8.00 s

Evaluating the expression:

α ≈ 9.07 rad/s²

Therefore, the constant angular acceleration of the wheel is approximately 9.07 rad/s².

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