Final answer:
The constant angular acceleration of the wheel can be calculated using kinematic equations for angular motion by converting revolutions to radians and then using the given final angular velocity and time to solve for the initial angular velocity and angular acceleration.
Step-by-step explanation:
To calculate the constant angular acceleration of the wheel, we can use the kinematic equation for angular motion which is analogous to the linear motion equation:
θ = ω_0 * t + (1/2) * α * t^2
Where θ is the angular displacement in radians, ω_0 is the initial angular velocity, α is the angular acceleration, and t is the time.
Given that the wheel rotates through 37.0 revolutions, we can convert this to radians by multiplying by 2π, since there are 2π radians in one revolution:
θ = 37.0 rev * 2π rad/rev = 232.4 rad
Now, we need another kinematic equation to relate the final angular velocity, initial angular velocity, angular acceleration, and time. The formula is:
ω = ω_0 + α * t
Given that the final angular velocity (ω) is 97.1 rad/s and the time (t) is 2.94 s, we can rearrange the second equation to solve for the initial angular velocity (ω_0) and then determine α using the first equation.
ω_0 = ω - α * t
With the above equations, we can set up a system to calculate α:
232.4 rad = ω_0 * 2.94 s + (1/2) * α * (2.94 s)^2
ω_0 = 97.1 rad/s - α * 2.94 s
However, we were not provided with the initial angular velocity, only the final velocity and total rotations. But since we know the final angular velocity and the time, we can calculate the initial velocity assuming constant acceleration:
ω_0 = ω - α * t
Now we substitute the first equation into the second:
232.4 rad = (97.1 rad/s - α * 2.94 s) * 2.94 s + (1/2) * α * (2.94 s)^2
This is a solvable equation for α, which is the angular acceleration of the wheel.