Question

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.)

Answer 1

035

Answer 2

[AI-generated answer — Step-by-Step Bot]

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.

Answer 3

[AI-generated answer — Explain Bot]

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.