A lathe, initially at rest, accelerates at 0.60 rad/s2 for 10 s, then runs at a constant angular velocity for 20 s, and finally decelerates uniformly for 10 s to come to a complete stop. What is its average angular velocity?

Question

A lathe, initially at rest, accelerates at 0.60 rad/s2 for 10 s, then runs at a constant angular velocity for 20 s, and finally decelerates uniformly for 10 s to come to a complete stop. What is its average angular velocity?

Answer 1

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

First, let's find the angular displacement during each phase of motion.

Phase 1: Acceleration
Using the formula for angular displacement during constant acceleration:
θ = ω_i * t + (1/2) * α * t^2
Where:
θ = angular displacement
ω_i = initial angular velocity
t = time
α = angular acceleration

Since the lathe is initially at rest, the initial angular velocity (ω_i) is 0 rad/s.
The time (t) for this phase is 10 s.
The angular acceleration (α) is given as 0.60 rad/s^2.

Plugging these values into the formula, we get:
θ1 = 0 * 10 + (1/2) * 0.60 * (10^2)
θ1 = (1/2) * 0.60 * 100
θ1 = 30 rad

Phase 2: Constant Velocity
During this phase, the angular displacement is constant since the angular velocity does not change. The angular velocity in this phase is not given.

Phase 3: Deceleration
Using the same formula as in phase 1, we can find the angular displacement during deceleration.
The initial angular velocity (ω_i) is the angular velocity at the end of the constant velocity phase.
The time (t) for this phase is 10 s.
The angular acceleration (α) is negative because it is decelerating and we assume it has the same magnitude.

Since the angular velocity is constant during the second phase, its value is equal to the final angular velocity in the first phase (v_f).
Therefore, v_f = α * t
v_f = 0.60 * 10
v_f = 6 rad/s

Now we can calculate the angular displacement during deceleration:
θ3 = v_f * t + (1/2) * α * t^2
θ3 = 6 * 10 + (1/2) * (-0.60) * (10^2)
θ3 = 60 - 30
θ3 = 30 rad

Next, let's calculate the total angular displacement:
Total angular displacement (θ_total) = θ1 + θ2 + θ3
θ_total = 30 + θ2 + 30
θ_total = 60 + θ2

Now we can calculate the average angular velocity:
Average angular velocity = θ_total / total time
Total time = 10 + 20 + 10
Total time = 40 s

Average angular velocity = (60 + θ2) / 40

Since we do not have the value of θ2, we cannot calculate the exact average angular velocity without more information. We would need to know the angular velocity during the constant velocity phase to complete the calculation.