Final answer:
To find the initial angular velocity of the flywheel, we need to calculate the angular acceleration using the final velocity, the time interval, and the total angle turned. With the angular acceleration, we can then determine the initial angular velocity from the start of the interval.
Step-by-step explanation:
The student is asking about the initial angular velocity of a flywheel at the beginning of a 6.0 s interval, given that it turns through 500 radians and attains an angular velocity of 100 rad/s within that time. Assuming the angular acceleration is constant, we can use the formula for angular motion, ω = ω_0 + αt, where ω is the final angular velocity, ω_0 is the initial angular velocity, α is the angular acceleration, and t is the time. Substituting the known values and solving for ω_0 will give us the initial angular velocity.
To find the angular acceleration, we use α = (ω - ω_0) / t. Once α is found, we substitute it and the final angular velocity (ω = 100 rad/s) back into the first equation to find ω_0. Let's perform the calculation:
The change in angular velocity is Δω = 100 rad/s – ω_0, and the time interval is Δt = 6.0 s. We know the flywheel turns through an angle of Δθ = 500 radians, which gives us another equation from the relationship Δθ = ω_0Δt + ½αΔt². By solving these equations concurrently, we can find the initial angular velocity and the angular acceleration. After simplifying and rearranging the equation, we will be able to determine ω_0.