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?

V1 = a*t1 = 0.6rad/s^2 * 10s. = 6 rad/s.

d1 = 0.5a*t1^2 = 0.3*10^2 = 30 Radians.

d2=V1*t2 = 6rad/s. * 20s. = 120 Rad1ans

V2 = V1 + a*t3 = 0
6 + a*10 = 0
10a = -6
a = -0.60 m/s^2

V2^2 = V1^2 + 2a*d = 0
d2 = -(V1^2)/2a = -(6^2)/-1.2 = 30 Rad.

V=(d1+d2)/(t1+t2+t3)=(120+30)/(10+20+10)
= 3.75 Rad/s.=Average angular velocity.

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

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

1. Acceleration phase:
The formula for angular displacement during constant acceleration is given by:
θ = ω0 * t + (1/2) * α * t^2
where θ is the angular displacement, ω0 is the initial angular velocity, α is the angular acceleration, and t is the time.

In this case, the initial angular velocity is 0 rad/s, the angular acceleration is 0.60 rad/s^2, and the time is 10 s.
θ₁ = (0) * (10) + (1/2) * (0.60) * (10)^2 = 30 rad

2. Constant angular velocity phase:
During this phase, the angular velocity remains constant, so there is no change in angular displacement.

θ₂ = 0 rad

3. Deceleration phase:
In this phase, the angular velocity reduces uniformly until the lathe comes to a complete stop. We can use the same formula as in the acceleration phase, but with a negative angular acceleration because it's decelerating.

In this case, the initial angular velocity is the same as the constant angular velocity during the second phase, which is also 0.60 rad/s. The angular acceleration is -0.60 rad/s^2, and the time is 10 s.
θ₃ = (0.60) * (10) + (1/2) * (-0.60) * (10)^2 = -30 rad

Now, let's calculate the total angular displacement:
Total angular displacement (θ) = θ₁ + θ₂ + θ₃
= 30 + 0 + (-30)
= 0 rad

Next, let's calculate the total time taken:
Total time taken = Time taken for acceleration phase + Time taken for constant velocity phase + Time taken for deceleration phase
= 10 s + 20 s + 10 s
= 40 s

Finally, we can calculate the average angular velocity:
Average angular velocity = Total angular displacement / Total time taken
= 0 rad / 40 s
= 0 rad/s

Therefore, the average angular velocity of the lathe is 0 rad/s.

To find the average angular velocity of the lathe, we need to calculate the total angular displacement it undergoes during the entire process.

First, let's find the angular displacement during the accelerated motion phase. We can use the formula:

θ = ω₀t + (1/2)αt²

Where:
θ is the angular displacement,
ω₀ is the initial angular velocity (which is zero in this case),
α is the angular acceleration,
t is the time.

Given that α = 0.60 rad/s² and t = 10s, we can plug in the values and find the angular displacement during the acceleration phase:

θ₁ = (1/2)(0.6)(10²)
θ₁ = 30 radians

Next, during the constant angular velocity phase, the angular displacement is simply the product of the angular velocity (which is constant) and the time t. Given the constant angular velocity for this phase, we will call it ω₂ and t = 20s:

θ₂ = ω₂t

For the deceleration phase, we can use the same formula as the acceleration phase since deceleration is just negative acceleration. The final angular velocity of the lathe will be zero, so the initial angular velocity ω₀₂ can be found by rearranging the formula:

ω = ω₀₂ + αt

Rearrange the formula to solve for ω₀₂:

ω₀₂ = -αt

Given that α = 0.60 rad/s² and t = 10s, we can find:

ω₀₂ = -(0.60)(10) = -6 rad/s

Now, using the formula for angular displacement during the deceleration phase:

θ₃ = ω₀₂t + (1/2)αt²

θ₃ = (-6)(10) + (1/2)(0.60)(10²)
θ₃ = -60 + 30
θ₃ = -30 radians

To calculate the total angular displacement, we add the displacements from each phase:

Total angular displacement, θ_total = θ₁ + θ₂ + θ₃

θ_total = 30 + (ω₂)(20) + (-30)

Since we want to find the average angular velocity, we divide the total angular displacement by the total time:

Average angular velocity = θ_total / total time

The total time is the sum of the acceleration time (10s), constant velocity time (20s), and deceleration time (10s):

Total time, t_total = 10 + 20 + 10 = 40s

Substituting the values into the formula:

Average angular velocity = θ_total / t_total

Finally, we can calculate the average angular velocity.