A lathe, initially at rest, accelerates at .60 rad/s^2 for 10s, then runs at a constant angular velocity for 20s, and finally decelerates uniformly for 10s to come to a complete stop. What is its average angular velocity?

Question

A lathe, initially at rest, accelerates at .60 rad/s^2 for 10s, then runs at a constant angular velocity for 20s, and finally decelerates uniformly for 10s to come to a complete stop. What is its average angular velocity?

Answer 1

See previous post: Sun,11-23-14, 12:08 AM.

Answer 2

To find the average angular velocity, we need to find the total angular displacement and divide it by the total time.

First, let's find the angular displacement during each phase:

1. During acceleration: To find the displacement, we can use the equation for angular displacement:
θ = ω_i * t + (1/2) * α * t^2
Here, ω_i is the initial angular velocity (0 rad/s), α is the acceleration (0.60 rad/s^2), and t is the time (10s).

Plugging in the values, we get:
θ_1 = (0 rad/s) * (10s) + (1/2) * (0.60 rad/s^2) * (10s)^2
θ_1 = 0 + 30 rad

So, the angular displacement during acceleration is 30 rad.

2. During constant angular velocity: Since the angular velocity remains constant, the angular displacement is simply given by the formula:
θ_2 = ω * t

Here, ω is the constant angular velocity and t is the time (20s).
Since the angular velocity is constant, we can use the final angular velocity, as it stays the same throughout.

θ_2 = ω_f * t

Given that the final angular velocity is not mentioned in the question, we cannot find the angular displacement during this phase.

3. During deceleration: Similar to the acceleration phase, we can use the same equation for angular displacement, but with negative acceleration this time.
θ = ω_i * t + (1/2) * α * t^2
Here, ω_i is the angular velocity at the end of the constant angular velocity phase, α is the acceleration (-0.60 rad/s^2), and t is the time (10s).

Plugging in the values, we get:
θ_3 = ω_f * (10s) + (1/2) * (-0.60 rad/s^2) * (10s)^2
θ_3 = 10ω_f - 30 rad

So, the angular displacement during deceleration is 10ω_f - 30 rad.

Now, let's calculate the total angular displacement by summing up the angular displacements during each phase:
Total angular displacement (θ_total) = θ_1 + θ_2 + θ_3

Since we don't have the value for θ_2, we can't calculate the total angular displacement or the average angular velocity accurately. We would need the given constant angular velocity during the second phase to proceed with the calculation.