Angular Speed/ Acceleration Problem

[~Sam~]

Homework Statement

Wind turbines is used to generate electricity. The length of each blade is r = 52.0 m. At t = 0, the rotor begins to turn counterclockwise, starting from rest. It speeds up at a constant rate for 60 s, until it reaches 12 revolutions per minute. Later on, it slows down from 12 rpm at a constant rate for 45 s, until it comes to a complete stop.

(a) While the turbine is speeding up, calculate its angular velocity ω⃗ and its angular ac- celeration α⃗ as functions of time. Calculate also the velocity ⃗v of the blade tip A as a function of time. These are all vector quantities, so to “calculate” means to give the components in certain directions. For the angular quantities, use the xyz system shown in the figure. For the quantities related to the blade tip, use the radial and tangential unit vectors rˆ and tˆ.

(b) How many revolutions does the turbine complete during the first 60 seconds while it is speeding up?

(c) Repeat the calculations of the previous part for the time when the blades are spinning down to a stop.

Homework Equations

I have done problems like these, but never with acceleration and deceleration.

The Attempt at a Solution

I'm kind of confused and trying to figure out all the jargon. I have tried to use the equation for angular speed and just subbing the delta change in angle with the delta time to account for acceleration. Then with that I could use it to find angular acceleration. I'm not particular sure if I'm in the right track at all. I'm also wondering if there's a way to parts of it with derivatives/antiderivatives.

 

[tiny-tim]

Welcome to PF!

Hi Sam! Welcome to PF!

(have a theta: θ and an omega: ω and an alpha: α )

You can use the usual constant acceleration equations (v_f = v_i + at etc), but with θ ω and α instead of s v and a.

 

[~Sam~]

Hi Sam! Welcome to PF!

(have a theta: θ and an omega: ω and an alpha: α )

You can use the usual constant acceleration equations (v_f = v_i + at etc), but with θ ω and α instead of s v and a.

Oh I see, thanks for the link. But for the angular equations, does t mean time? Or time it takes to complete 1 rev (the period).

 

[tiny-tim]

In the constant acceleration equations, t always means time between initial and final.

 

[~Sam~]

In the constant acceleration equations, t always means time between initial and final.

So in this case for angular velocity/acceleration, t would just be the time interval we want where the velocity/acceleration is occurring.