Can you help me explain how to find angular velocity?

[mysteryowl]

A machinist turns the power on to a grinding wheel, at rest, at time t = 0 s. The wheel accelerates uniformly for 10 s and reaches the operating angular velocity of 69 rad/s. The wheel is run at that angular velocity for 33 s and then power is shut off. The wheel decelerates uniformly at 2.9 rad/s2 until the wheel stops. In this situation, the average angular velocity in the time interval from t = 0 s to t = 25 s is closest to:

I know that 1st there is constant acceleration, then constance velocity, and then constant deacceleration.

im really confused, the answer is 55 rad/s, but i don't know how to get the answer

 

[mfb]

Did you draw a sketch of the angular velocity as function of t?
If you have such a sketch, how can you get the average value between two points?

 

[Gavran]

ω_avg=Δθ/Δt
ω_avg=(Δθ₁+Δθ₂)/(Δt₁+Δt₂)
ω_avg=[(θ_t=10s-θ_t=0s)+(θ_t=25s-θ_t=10s)]/[(t_t=10s-t_t=0s)+(t_t=25s-t_t=10s)]

 

[jbriggs444]

ω_avg=Δθ/Δt
ω_avg=(Δθ₁+Δθ₂)/(Δt₁+Δt₂)
ω_avg=[(θ_t=10s-θ_t=0s)+(θ_t=25s-θ_t=10s)]/[(t_t=10s-t_t=0s)+(t_t=25s-t_t=10s)]

You did the average over just two time intervals. There are three time intervals in the problem.

Your next problem is to compute $\Delta \theta$ for each of the three intervals. What are your thoughts about the rotation angle for the first interval (the acceleration) alone?

 

[Steve4Physics]

It may be worth noting that the original post is over 10 years old!

 

[kuruman]

It may be worth noting that the original post is over 10 years old!

That it is. Additional replies are unlikely to help @mysteryowl who hasn't been seen since that first post. However, it's not too late to guide @Gavran along the path to the correct answer.

 

[Gavran]

You did the average over just two time intervals. There are three time intervals in the problem.

Your next problem is to compute $\Delta \theta$ for each of the three intervals. What are your thoughts about the rotation angle for the first interval (the acceleration) alone?

The task is to calculate the average velocity in the time interval from $t=0s$ to $t=25s$. This does not include the third (deceleration) interval from $t=10s+33s=43s$ to until the wheel stops.
The rotation angle for the first interval is:
$\Delta\theta_1=\frac12\frac{\Delta\omega_1}{\Delta t_1}(\Delta t_1)^2=\frac12\Delta\omega_1\Delta t_1$
where $\Delta\omega_1$ is the angular velocity change in the first interval and $\Delta t_1$ is the duration of the first interval.