An antique carousel that’s powered by a large electric motor undergoes constant angular acceleration from rest to full rotational speed in 6 seconds. When the ride ends, a brake causes it to decelerate steadily from full rotational speed to rest in 8.9 seconds. Compare the torque that starts the carousel to the torque that stops it.
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To compare the torque that starts the carousel to the torque that stops it, we need to analyze the rotational motion happening during the acceleration and deceleration phases. Let's break it down step by step:
Step 1: Calculate the angular acceleration during the acceleration phase.
Given that the carousel goes from rest to full rotational speed in 6 seconds, we can find the angular acceleration using the formula:
angular acceleration (α) = (final angular velocity (ω) - initial angular velocity (ω₀)) / time (t)
Since the carousel starts from rest, the initial angular velocity (ω₀) is 0. We can assume that the final angular velocity (ω) is the full rotational speed. Let's denote the final angular velocity as ω₁.
So, α = (ω₁ - 0) / 6
Step 2: Calculate the torque that starts the carousel.
The torque (τ) can be calculated using the formula:
torque (τ) = moment of inertia (I) * angular acceleration (α)
We need information about the moment of inertia to proceed further.
Step 3: Calculate the angular deceleration during the deceleration phase.
Given that the carousel decelerates steadily from full rotational speed to rest in 8.9 seconds, we can find the angular deceleration using a similar formula:
angular deceleration (α') = (initial angular velocity (ω₁) - final angular velocity (0)) / time (t')
Here, the initial angular velocity (ω₁) is the full rotational speed, and the final angular velocity is 0. Let's denote the angular deceleration as α'.
Step 4: Calculate the torque that stops the carousel.
Using the same formula as before:
torque (τ') = moment of inertia (I) * angular deceleration (α')
In order to fully analyze and compare the torques, we require the moment of inertia (I) for the carousel. The moment of inertia depends on the mass distribution of the carousel and its shape.
Please provide information about the mass distribution or shape of the carousel so that we can proceed with the calculations.