While at the county fair you take a ride on the carousel and notice that it goes around once in 12 s. The radius of the carousel is 15 m. As the ride comes to an end, it slows down smoothly, coming to a stop in 2.5 revolutions. What is the magnitude of the rotational acceleration of the carousel while it is slowing down?
To find the magnitude of the rotational acceleration of the carousel, you need to calculate the angular acceleration first. Angular acceleration (α) is defined as the rate of change of angular velocity (ω), which in turn is the rate of change of angle (θ) with respect to time (t).
1. First, calculate the initial angular velocity (ω0) of the carousel using the information given. The time for one revolution is 12 seconds, which means the frequency (f) is the reciprocal of the time: f = 1/12 s^(-1). The angular velocity (ω) can be calculated using the formula ω = 2πf.
ω0 = 2πf = 2π(1/12 s^(-1)) = π/6 rad/s
2. Next, calculate the final angular velocity (ωf) of the carousel when it comes to a stop. The carousel comes to a stop in 2.5 revolutions, which corresponds to an angle of 2.5 * 2π radians. The time taken for this to happen is unknown, so let's call it tf.
ωf = Change in angle / Time taken = (2.5 * 2π) rad / tf = 5π rad / tf
3. Now, use the formula for angular acceleration: α = (ωf - ω0) / t. Rearrange the formula to solve for α:
α = (ωf - ω0) / tf
Substituting the values we found earlier:
α = (5π rad / tf - π/6 rad/s) / tf
Simplifying the expression:
α = (5π - π/6) / tf
α = (30π - π) / (6tf)
α = 29π / (6tf) rad/s^2
Therefore, the magnitude of the rotational acceleration of the carousel while slowing down is 29π / (6tf) rad/s^2, where tf is the time taken for the carousel to come to a stop in 2.5 revolutions.
To find the magnitude of the rotational acceleration of the carousel, we need to calculate the angular acceleration.
First, let's find the angular velocity (ω) of the carousel when it comes to a stop. We can use the equation:
ω = (2π * N) / T
where ω is the angular velocity, N is the number of revolutions, and T is the time.
Given that the carousel comes to a stop in 2.5 revolutions and takes 12 seconds to complete one revolution, we can plug the values into the equation:
ω = (2π * 2.5) / 12
Simplifying this equation, we get:
ω = (5π) / 12
Next, let's find the initial angular velocity (ω₀) of the carousel. We know that the carousel's angular velocity is constant throughout, so we can use the initial and final angular velocities to find the average angular velocity, and then use the equation:
Δω = α * Δt
where Δω is the change in angular velocity, α is the angular acceleration, and Δt is the change in time.
Given that the carousel starts from rest and ends at ω, we have:
Δω = ω - ω₀
Since ω₀ is 0 (starts from rest), we can rewrite the equation as:
α = Δω / Δt
Substituting the values, we get:
α = ((5π) / 12) / 2.5
Simplifying this equation, we find:
α ≈ 0.628 m/s²
Therefore, the magnitude of the rotational acceleration of the carousel while it is slowing down is approximately 0.628 m/s².